TITLE: Simplification of complex exponentiation of a number
QUESTION [0 upvotes]: Can someone help me find the simplification (like $a+bi$) to the expression $x^i$, where $x$ is a number? How do I proceed?
For example.. what are the $a$ and $b$ of the number $5^i$?
Thank you :)

REPLY [0 votes]: This will always be multiple-valued.
One generally defines $$z^w \stackrel{\small{\textrm{def}}}{=} e^{w\log z}$$
and the multiple-valuedness of the logarithm produces multiple values for this expression.
If you write $z=re^{it}$ where $r>0$, and $w=u+iv$ with $u$ and $v$ real, then the possible values are precisely (where "$\ln$" is the real-valued function of a real variable)
$$e^{(u+iv)(\ln r + i(t + 2k\pi))}$$
$$=e^{u\ln r - v(t+2k\pi) + i(v\ln r + u(t+2k\pi))}$$
$$=r^ue^{-v(t+2k\pi)}\cdot e^{i(\ln r^v + u(t+2k\pi))}$$
$$=r^ue^{-v(t+2k\pi)}\cdot e^{i\ln (r^v e^{ u(t+2k\pi)})}$$
$$=\boxed{r^ue^{-v(t+2k\pi)}\cos\ln(r^v e^{ u(t+2k\pi)}) + ir^ue^{-v(t+2k\pi)}\sin\ln(r^v e^{ u(t+2k\pi)})}$$
for integral $k$.

Note: Taking $k=0$ yields a somewhat simpler "principal" value
$$\boxed{r^ue^{-vt}\cos\ln(r^v e^{ ut}) + ir^ue^{-vt}\sin\ln(r^v e^{ ut})}$$
For your example, we write $z=5=5e^{0i}$, so $r=5$, $t=0$, $u=0$, and $v=1$. This gives the possible values
$$=\boxed{e^{-2k\pi}\cos\ln 5 + ie^{-2k\pi}\sin\ln 5}$$
with "principal" value (taking $k=0$)
$$=\boxed{\cos\ln 5 + i\sin\ln 5}$$


TITLE: Prove that the space is totally-disconnected.
QUESTION [2 upvotes]: Given.
$$D \text{ - some dense set in ℝ} $$
$$P(D) - \text{power set over D} $$
$$T_{d_E} - \text{canonical topology on ℝ} $$
$$T_D = T_{d_E}\cup P(D) $$
$$T - \text{ topology generated from the subbasis } T_D$$
$$\underline X = (ℝ, T) $$
Prove.
$\underline X$ is totally-disconnected.

Attempt.
If $A \subseteq R, \ card(A) > 1$ and $A$ is open, then it contains some $d \in D$ by the definition of density.
Since $\underline X$ is a $T_1$ space, then $\{d\}$ is closed but it's also open in $A$ due to the way $T_D$ is defined. So $\{d\} \neq A$ is "clopen" set in $A$ and hence $A$ is disconnected.
Same goes by the above logic for the case if $A$ is not open but is a neighbourhood containing an open set.
If the above is right then one can handle the cases: $[ a, b], $[a, b), $(a, b]$ and $(a,b)$.
But how can one show that e.g. non-interval set is disconnected: $B = (a,b)\cup(c,d)$?
If the topology defined on $\underline X$ were $T(\le)$ Topology, then one could argue that since $B$ is not an interval then it can't be connected. But $T_D$ is not an order topology, so how can I prove that $B$ type sets are disconnected too?

REPLY [1 votes]: $T_D$ is not a topology but it is a subbase for a topology, which is probably what's meant here. Taking $D= \Bbb P$ gives us the Michael line, and this question is meant to generalise this situation, I suppose.
To see that $\underline{X}$ is totally disconnected, take a subspace $A$ having $2$ or more points. If $d \in D \cap A \neq \emptyset$ then $\{d\}$ is a clopen subset of $A$, so $A$ is not connected. Otherwise $A$ has empty interior in the Euclidean topology and $A \subseteq \Bbb R\setminus D$. In the usual topology any $A$ that has empty interior and has two or more points is disconnected (as it's not an interval of any type) and so the same holds for $A$ (which has the Euclidean topology, been disjoint from $D$). QED


TITLE: Chebyshev's Inequality proof explanation - squared values of probability
QUESTION [0 upvotes]: In the proof of Chebyshev's Inequality we do the following:
Claim: for some random variable $Y$ and some value $a > 0$
$Pr[|Y - E[Y]| \geq a] = \frac{E[(Y - E[Y])^2]}{a^2}$
Let's refer to $E[Y]$ as $\mu$
\begin{align*}
Pr[|Y - \mu| \geq a] =&\  Pr[|Y - \mu|^2 \geq a^2] \\
=& \ Pr[X \geq a^2]\ \ \ \ if \ X = (Y - \mu)^2\\
\leq& \ \frac{E[X]}{a^2}\ \ \ \ by \ Markov's \ inequality\\
=& \ \frac{E[(Y - E[Y])^2]}{a^2}
\end{align*}
My question is, how can we just assert:
$Pr[|Y - \mu| \geq a] =\  Pr[|Y - \mu|^2 \geq a^2]$
What is the intuition or line of thought behind this?

REPLY [1 votes]: Another way to see it is that
$$\mathbb{P}(|X|\geq a)=\mathbb{E}\left[1_{|X|\geq a}\right]$$
Now look at the indicator function:
$$1_{|X|\geq a}=\begin{cases}
1 & \text{if }|X|\geq a\\
0 & \text{if}\,|X|<a
\end{cases}=\begin{cases}
1 & \text{if }|X|^{2}\geq a^{2}\\
0 & \text{if}\,|X|^{2}<a^{2}
\end{cases}$$
As it preserves inequalities in this setting. Meaning: $1_{|X|\geq a}=1_{|X|^{2}\geq a^{2}}$. It is the same analogy as Epiousios, but just specifying that the indicator function is equal in both cases.


text:
\begin{document}

\title{Matheme and mathematics. On the main concepts of the philosophy of Alain Badiou.}
\author{Maciej Malicki \\
Department of Mathematics and Mathematical Economics, \\
Warsaw School of Economics, \\

mamalicki@gmail.com}
\date{Jan 1, 2014}

\maketitle

\begin{abstract}
In this paper, I present a critical discussion of the mathematical arguments employed in the philosophy of event of Alain Badiou.  On the basis of ``Being and Event'' as well as his other writings, I analyze the main notions of his philosophy such as the indiscernible, the undecidable, and the unnameable. The focus of my analysis is both on their mathematical consistency, and their philosophical consequences. I argue that the mathematical approach developed by Badiou is seriously defective, and, as a result, that it cannot serve as an ontological basis for the concept of event as presented in ``Being and Event''.
\end{abstract}

\section{Introduction}

``Being and event'' is an ontological theory of the effect of event, it is a science of the possibility of a novelty that breaks with being's determinants. Its existence can never be decided once and for all, because the occurrence of an event is always an intervention: its ultimate source lies in a decision of the subject who is constituted in this act. As a result, a post-evental truth enters being. It is infinite and indiscernible, which means that it cannot be described in terms of authoritative knowledge. It modifies the situation by overruling some elements of knowledge, and making other veridical.
Even though a truth cannot be accessed in its entirety, the subject supporting a procedure of truth can decide that something belongs to its realm. These gestures of fidelity give rise to the subject language, which forces statements describing the state of affairs after the occurrence of an event.

These are the core concepts of Badiou's philosophy of event in a nutshell. However ---  and that is perhaps the most striking feature of ``Being and event'' ---  all of its fundamental categories have a mathematical character, and main theses of this book derive from subtle and quite technical considerations based on the area of mathematics called set theory. Eventually, the science of event turns out to be the result of an investigation into set theory, understood as ontology, or rather proved to be ontology, i.e. the theory of ``being qua being''.

In this paper, I am mostly concerned with consistency of mathematical aspects of Badiou's analysis, as well as some of its philosophical consequences in certain strictly defined key aspects. It is an important reservation, as a large part of Badiou's thought can be presented without referring to any technical terms. Many of its categories ---  such as event itself, subject, fidelity, the void or the excess --- are well rooted in the language of contemporary philosophy.

Taking a broader perspective, I am interested in a postulated deep ontological structure behind four fundamental domains of truth which are science, politics, art and love.  Of course, we do not need any special doctrine in order to recognize a groundbreaking character of basic examples of event discussed by Badiou --- Mallarme's poetry, relativistic physics, the communist revolution or dodecaphony, as well as some similarities between them --- a break involved in their occurrence, a resulting new paradigm, a peculiar indeterminateness of their status. On the other hand, some of these examples do not seem to have that much in common with others, at least on the surface of things. In order to acknowledge the relevance of Badiou's position, more is needed than a simple rephrasing of well known ideas in a new language: what is required is a convincing argument showing that the ontological theory of event is an autonomous interpretative tool, independent of specific subject matters. Moreover, it is rather hard to imagine an argument of this kind that would not contain a proof that the mathematical part of Badiou's thought is consistent, and it agrees with his fundamental philosophical aims. A formalized doctrine presented by Badiou not only allows for a realization of this postulate but it explicitly requires it.

Finally, one could describe this article as a contribution to the reflection on the concept of matheme. In ``Being and Event'', the matheme is understood as a philosophical idea subjected to rigors of deduction, and opposed to the pre-platonic poem. However, the very term `matheme' comes from late writings of Lacan, which are an important reference point for Badiou.  There it denotes mathematical objects  ---  such as the Boromean knot or the Klein bottle ---  allowing to grasp the structure of the order of the real which extends beyond the reach of language. Under such understanding, a matheme would be a place in philosophy, where mathematics attains an autonomous status, completing philosophical discourse and generating statements that are binding for it.

My assessment of the method employed in ``Being and Event'' is negative.  After providing the first part of Badiou's doctrine, that is the identification of ontology with set theory, and outlining mathematical foundations of the theory of event, I present a critical discussion of four key concepts of this theory: the indiscernible, the undecidable, the unnameable, and the evental site. The conclusion is that their mathematical structure and its implications for Badiou's philosophy turn out not to meet expectations. They are ill-designed, and this leads to mathematical inconsistencies as well as philosophical consequences that contradict Badiou's intentions.  Far from rejecting the philosophical substance of these categories, I claim that their mathematical content  may serve as an inspiring metaphor or analogy, but it has clearly defined bounds of meaningful interpretation.

\section{Mathematics as ontology}

The edifice of Badiou's philosophy of ``being qua being'' is founded on a thesis establishing the identity of mathematics and ontology. In order to accomplish this task, Badiou in the very first words of ``Being and Event'' goes back to the old Parmenedean dispute on the status of the one and the multiple. He decides it by giving priority to the multiple, while considering the one as an effect of operation of imposing a structure upon being, organizing the 'primordial' \emph{inconsistent multiplicity}. Inconsistent means here exactly this: ``without any unifying principle''.

Being such as it presents itself ---  and ``no access to being is offered to us except presentations'' \cite[p. 27]{BE} ---   is a \emph{consistent multiplicity}. It is a multiplicity because ``if the one is not reciprocal with being, the multiple, however, is reciprocal with presentation.'' \cite[s. 28]{BE}; also, it is consistent because a presented being is always the result of a certain organizing principle, certain \emph{operation of counting-as-one}. The inconsistent multiplicity is a \emph{subtractive} basis of an already structured presentation --- it is possible to be discerned only retroactively, as that `something' on which a count-as-one operated \cite[p. 25]{BE}.  From the perspective of presentation it is only the void because the effect of structure encompasses everything without exception. In this manner, the inconsistent multiplicity ``sutures presentation to being'' \cite[p. 55]{BE}; the void is the name of being \cite[p. 56]{BE}. This establishment of a consistent multiplicity resulting from an operation of counting-as-one, a combination of the earlier inconsistent and later consistent multiplicity is called a \emph{situation}.

It is the inconsistent multiplicity that forms the domain of ontology. ``Ontology can be solely the theory of inconsistent multiplicities as such. 'As such' means that what is presented in the ontological situation is the multiple without any other predicate than its multiplicity.''  \cite[p. 28]{BE} How is the presentation of being regarded as multiplicity --- the presentation of presentation --- possible if ``being has no structure'' (s. 27)? What requirements does such a situation need to satisfy? Firstly, the multiple without one is the multiple that consists only of multiplicities. Secondly, the operation of counting-as-one of the ontological situation cannot be anything more than a collection of conditions through which a multiple can be recognized as multiple. Finally, ontology must be a theory of ``the suture of presentation to being'', that is a theory of the subtractive void of presentation. In other words, it should derive the existence of its multiplicities only out of the void.

The science satisfying these postulates is set theory --- a mathematical theory of the relation of belonging $\in$, together with appropriate axioms: axiom of extensionality, regularity, pairing, union, infinity, power set, and axiom schema of comprehension and replacement. First of all, the universe of set theory contains only sets, so set theory presents only multiplicities consisting of multiplicities. Moreover, set theory is an axiomatic theory, and axioms are formal rules that do not explicitly define objects they refer to. Badiou says: ``an axiomatic presentation consists, on the basis of non-defined terms, in prescribing the rule for their manipulation.'' \cite[p. 29]{BE} Sets \emph{de facto} satisfy axioms of set theory, however the axioms themselves do not form a criterion of being a set. In other words, what axioms determine about sets is that they are multiplicities, nothing more. Lastly, the unpresented subtractive void of situation appears in set theory in the form of the empty set, which is a basic building block of all sets. It must be a set because every element presented in the ontological situation is the effect of its operation of counting-as-one, however in fact it is a ``'multiple' which is neither one nor multiple, being the multiple of nothing, and therefore, as far as it is concerned, presenting nothing in the form of the multiple, no more than in the form of the one. This way ontology states that presentation is certainly multiple, but that the being of presentation, the that which is presented, being void, is subtracted from the one/multiple dialectic.'' \cite[p. 59]{BE}

\section{Belonging, inclusion, and the impasse of being}

Let us take a closer look at the mathematical structure of key categories of ``Being and Event''. Ontologically, situations --- as it has been already said --- take the form of sets: multiplicities whose one is nothing more than the unity of the elements they consists of. Two basic types of relations between sets are \emph{belonging}, that is being an element of a set, being \emph{presented in a situation}, and \emph{inclusion}, that is, being a subset of a set, or being \emph{represented in a situation}. Even though the latter relation is formally reducible to the former one --- $y$ is a subset of $x$ if and only if every element of $y$ is an element of $x$ --- their properties differ considerably The tension between belonging and inclusion is of fundamental type in so far as it is where the ``impasse of being'' arises, opening up a situation to the interventional occurrence of an event. More light can be shed on this tension with the help of the concept of power set.

The power set $p(x)$ of $x$ --- \emph{the state of situation} $x$  or its \emph{metastructure} \cite[p. 94]{BE} --- is defined as the set of all subsets of $x$. Now, the basic relations between sets can be expressed as the following relations between sets and their power sets. If for some $x$, every element of $x$ is also a subset of $x$, then $x$ is a subset of $p(x)$, so $x$ can be reduced to its power set. Conversely, if every subset of $x$ is an element of $x$, then $p(x)$ is a subset of $x$, and the power set $p(x)$ can be reduced to $x$. Sets that satisfy the first condition are called \emph{transitive}. For obvious reasons the empty set is transitive; other examples of transitive sets also can be easily found. However, the second relation never holds. The mathematician Georg Cantor proved that not only $p(x)$ can never be a subset of $x$, but in some fundamental sense it is essentially larger than $x$. On the other hand, axioms of set theory do not determine the extent of this difference. Badiou says that it is an ``excess of being'', an excess that at the same time is its impasse.

In order to explain the mathematical sense of this statement, one needs to recall the notion of \emph{cardinality}, which clarifies and generalizes a common understanding what quantity is. We say that two sets $x$ and $y$ have the same cardinality if there exists a function defining a one-to-one correspondence between elements of $x$ and elements of $y$. For finite sets, this definition agrees with common intuitions: if a finite set $y$ has more elements than a finite set $x$ (say, $y$ has $10$ elements and $x$ has $7$ elements), then regardless of the method of assigning elements of $y$ to elements of $x$, something (some $3$ elements) will be left over in $y$ --- precisely because it is larger. In particular, if $y$ contains $x$ \emph{and} some other elements, then $y$ does not have the same cardinality as $x$. This seemingly trivial fact is not always true outside of the domain of finite sets. To give a simple example, the set of all natural numbers $\NN$ contains quadratic numbers, that is numbers of the form $n^2$, as well as some other numbers but these two sets have the same cardinality. A correspondence witnessing this fact assigns to every number $n$ a unique quadratic number, namely $n^2$.

Counting finite sets has been always done via natural numbers $0,1,2, \ldots$ . In set theory, the concept of such a canonical measure can be extended to all infinite sets using the notion of \emph{cardinal number}. Without getting into details of their definition, let us say that the series of cardinal numbers begins with natural numbers, which are directly followed by the number $\omega_0$ that measures the size of the set of all natural numbers $\NN$, then by $\omega_1$, the first uncountable cardinal numbers, etc. The hierarchy of cardinal numbers has the property that every set $x$, finite or infinite, has cardinality equal to exactly one cardinal number $\kappa$. We say then that $\kappa$ is the cardinality of $x$.

It is not hard to prove that the cardinality of the power set $p(x)$ is $2^n$ if the cardinality of $x$ is finite and equal to $n$. However, something seemingly quite paradoxical happens when infinite sets are taken into account.  Even though Cantor's theorem does state that  the cardinality of $p(x)$ is always larger than $x$ --- similarly as in case of finite sets --- axioms of set theory \emph{never} determine the exact cardinality of $p(x)$ of an infinite set $x$. Moreover, one can formally prove that there exists no proof determining the cardinality of the power sets of any infinite set. There is a general method of building models of set theory, discovered by the mathematician Paul Cohen, and called \emph{forcing} that yields models, where --- depending on construction details --- cardinalities of infinite power sets take different values. Consequently, quantity --- ``a fetish of objectivity'' \cite[p. 83]{BE} as Badiou calls it --- does not define a measure of being but it leads to its impasse instead. It reveals an undetermined gap \cite[p. 83]{BE}, where an event can occur --- ``that-which-is-not-being-qua-being'' \cite[p. 184]{BE}.

\section{Forcing, truth and a place of the subject}

In order to make the exposition more accessible, let us consider only the power set of the set $\NN$ of all natural numbers, which is the smallest infinite set, i.e. a countable infinity. Simplifying things slightly, the argument proceeds as follows. By a model of set theory, we understand a set, where, if we restrict ourselves to its elements only, all axioms of set theory are satisfied. It follows from G\"{o}del's completeness theorem that as long as set theory is consistent, no statement valid in a model of set theory can contradict logical consequences of its axioms. If the cardinality of $p(\NN)$ was a consequence of axioms of set theory, there would exist a cardinal number  $\kappa_1$ such that the sentence 'the cardinality of $p(\NN)$ is  $\kappa_1$' was true in all the models. However, for every cardinal $\kappa_1$ the technique of forcing allows for constructing a model $M$ where the cardinality of $p(\NN)$ is equal to some cardinal $\kappa_2$ that differs from $\kappa_1$. Then the sentence 'the cardinality of $p(\NN)$ is  $\kappa_2$' is true in $M$, while at the same time it contradicts the sentence 'the cardinality of $p(\NN)$ is  $\kappa_1$'. This is impossible assuming consistency of set theory, so for no $\kappa_1$  the sentence 'the cardinality of $p(\NN)$ is  $\kappa_1$' is a consequence of axioms of set theory. Therefore they do not decide the cardinality of $p(\NN)$.

The starting point of forcing is a model $M$ of set theory --- called the \emph{ground model} --- that is countably infinite and transitive (it is a crucial assumption). As a matter of fact, the existence of such a model cannot be proved but it is known that there exists a countable and transitive model for every \emph{finite} subset of axioms. As far as the logic of the construction is concerned, in particular decidability of the sentences obtained by forcing, this difference does not play any role.

A characteristic subtlety can be observed here. From the perspective of `an inhabitant of the universe', that is if all the sets are considered, the model $M$ is only a small part of this universe. It is deficient in almost every respect; for example all of its elements are countable, even though the existence of uncountable sets is a consequence of axioms of set theory. However, from the point of view of an `inhabitant of $M$', that is if elements outside of $M$ are disregarded, everything is in order. Some of the sets that are countable in the universe are actually uncountable in $M$ because in this model there are no functions establishing a one-to-one correspondence between them and $\omega_0$. One could say that $M$ simulates properties of the whole universe.

The main objective of forcing is to build a new model $M[G]$ based on $M$, which contains $M$, and satisfies certain additional properties. The model $M[G]$ is called the \emph{generic extension} of $M$. In order to accomplish this goal, a particular set is distinguished in $M$ --- its elements are referred to as conditions --- which will be used to determine basic properties of the generic extension. In case of the forcing that proves undecidability of the cardinality of $p(\NN)$, the set of conditions codes finite fragments of a function witnessing the correspondence between $p(\NN)$ and a fixed cardinal $\kappa_2$.

In the next step, an appropriately chosen set $G$ is added to $M$ as well as other sets that are indispensable in order for $M[G]$ to satisfy axioms of set theory. The set $G$ --- called \emph{generic} --- is a subset of the set of conditions and it always lays outside of $M$. The construction of $M[G]$ is exceptional in so far as its key properties can be described and proved using $M$ only, or even the set of conditions, thus without referring to the generic set. It is possible for three reasons. First of all, every element of $M[G]$ has a name existing already in $M$, and built using the conditions. Second of all, based on these names one can design a language called the \emph{forcing language} or ---  as Badiou terms it ---  the \emph{subject language} that is powerful enough to express every sentence of set theory referring to the generic extension. Finally, it turns out that validity of sentences of the forcing language in the extension $M[G]$ depends on the set of conditions: the conditions \emph{force} validity of sentences of the forcing language in a precisely specified sense. As it has been already said, the generic set $G$ consists of some of the conditions, so even though $G$ is outside of $M$, its elements are in $M$. Recognizing which of them will end up in $G$ is not possible for an inhabitant of $M$, however in some cases the following can be proved: provided that condition $p$ is an element of $G$, sentence $S$ is true in the generic extension constructed using this generic set $G$. We say then that $p$ forces $S$.

In this way, with the aid of the forcing language one can prove that every generic set of the Cohen forcing codes an entire function defining a one-to-one correspondence between elements of $p(\NN)$ and a fixed cardinal number --- it turns out that \emph{all} of the conditions force the sentence stating this property of $G$, so regardless of what conditions end up in the generic set, it is always true in the generic extension. On the other hand, the existence of a generic set in the model $M$ cannot follow from axioms of set theory, otherwise they would decide the cardinality of $p(\NN)$.

The method of forcing is of fundamental importance for Badiou's philosophy. The event escapes ontology; it is ``that-which-is-not-being-qua-being'', so it has no place in set theory or the forcing construction. However, the post-evental truth that enters and modifies the situation is presented by forcing in the form of a generic set that leads to an extension of the ground model. In other words, the situation understood as the ground model $M$ is transformed by a post-evental truth identified with a generic set $G$, and becomes the generic model $M[G]$. Moreover, the knowledge of the situation is interpreted as the language of set theory, serving to discern elements of the situation; and as axioms of set theory, deciding validity of statements about the situation. Knowledge understood in this way does not decide the existence of a generic set in the situation nor can it point its elements (this property of truth will be thoroughly discussed later). A generic set is always \emph{undecidable} and \emph{indiscernible}.

Therefore from the perspective of knowledge it is not possible to establish, whether a situation is still the ground-model or it has undergone a generic extension resulting from the occurrence of an event; only the subject can interventionally decide it.  And it is only the subject who decides about belonging of particular elements to the generic set. A \emph{procedure of truth} or \emph{procedure of fidelity} \cite[p. 329]{BE} supported in this way gives rise to the subject language. It consists of sentences of set theory, so in this respect it is a part of knowledge, nonetheless \emph{veridicity} of the subject language originates from decisions of the faithful subject. Consequently, a procedure of fidelity forces statements about the situation as it is after being extended and modified by the operation of truth.

\section{Mathemes of the undecidable and of the evental site}

According to Badiou, an undecidable truth is located beyond boundaries of authoritative claims of knowledge. At the same time, undecidability indicates that truth has a post-evental character: ``the heart of the truth is that the event in which it originates is undecidable'' \cite[p. 221]{BE}. Badiou explains that, in terms of forcing, undecidability means that the conditions belonging to the generic set force sentences that are not consequences of axioms of set theory. However, one also needs to answer the question about the role axioms play in the structure of \emph{historical} situations. If in the domains of specific languages (of politics, science, art or love) the effects of event are not visible, ``Being and Event'' is an empty exercise in abstraction: even science --- perhaps excluding some entirely formalized areas of theoretical physics --- let alone art or love  --- cannot for obvious reasons be exhaustively described in terms of the sole relation of belonging. Anyway, it is doubtful that --- to consider just one example --- the status of the French revolution is different from the status of the absolute monarchy preceding it as far as the axioms of set theory are concerned. In other words, most likely either all historical facts are decidable or none of them is. Both possibilities lead to a trivial notion of the undecidable.

Judging by numerous examples discussed by Badiou, it seems that he distances himself form such a narrow interpretation of the function played by axioms. He rather regards them as collections of basic convictions that organize situations, the conceptual or ideological framework of a historical situation. For example, the nature of politics in Rousseau's writings is formulated in the following way: ``The major axiom is that in order to definitely have the expression of the general will, [there must] be no partial society in the State'' \cite[p. 348]{BE}. This approach is also indicated in the only part of ``Being and Event'' which is devoted to that issue in general terms: ``Let us agree that a proposition is singular (\ldots) if, within a historically structured mathematical situation, it implies many other significant propositions, yet it cannot itself be deduced from the axioms which organize the situation. (\ldots) Say that $A$ is this proposition. (\ldots) An event, named by an intervention, is then, at the theoretical site indexed by the proposition $A$, a new apparatus, demonstrative or axiomatic, such that $A$ is henceforth clearly admissible as a proposition of the situation.'' \cite[p. 246]{BE} Accordingly, undecidability of a truth would consist in transcending the theoretical framework of a historical situation or even breaking with it in the sense that the faithful subject accepts beliefs that are impossible to reconcile with the old mode of thinking.

A clear illustration of the effect of event that in Badiou's opinion results in breaking with the determinants of the old paradigm, rather than just in moving beyond them, is the birth of  relativistic physics: ``After Einstein's texts of 1905, if I am faithful to their radical novelty, I cannot continue to practice physics within its classical framework'' \cite[p. 42]{E}. The novelty of  relativistic physics cannot be reduced to a mere substitution of one formulas with others, more precise, because it gives rise to a completely new understanding of fundamental physical categories such as space, time, reference point or motion. For profound reasons classical mechanics rules out --- instead of simply not deciding it --- the very possibility of emerging the theory of relativity within its own conceptual framework. Similarly, the French revolution and communism --- essential examples of the effect of event --- violently rupture the historical order preceding them. The French revolution overthrew the king and established sovereignty of the people. And if something can be said with certainty about any real or imaginary realization of the communist idea, it is that it definitely abolishes the ``capital-parliamentarism''.

However, if one consequently identifies the effect of event with the structure of the generic extension, they need to conclude that these historical situations are by no means the effects of event. This is because a crucial property of every generic extension is that axioms of set theory remain valid in it. It is the very core of the method of forcing, stated in the Theorem of the Generic Model \cite[Th. 14.5]{Jech}. Without this assumption, the original Cohen's construction would have no \emph{raison d'etre} because it would not establish undecidability of the cardinality of  infinite power sets. Let us say this once more: every generic extension satisfies axioms of set theory. In reference to historical situations, it must be conceded that a procedure of fidelity may modify a situation by forcing undecidable sentences, nonetheless it never overrules its organizing principles.

From the point of view of the generic theory of truth, some hypothetical type of social-democracy might be considered as the effect of event. It would abolish chaos and inequalities resulting from mechanisms of democratically controlled market economy by the operation of a new idea that would transfigure the nature of these mechanisms from within. As a religious event, transgressing the Law without literally breaching it, one could probably point to Messianic Judaism or Protestantism. Another interesting case is a theory of the Danish astronomer Tycho de Brahe who in XVI century proposed a solution that allowed for keeping empirical advantages of the heliocentric model, while letting the Earth stay in the center of the Universe. In this conception, all the planets revolve around the Sun, except for the Earth, which is encircled by it.  In terms of kinetics, that is if the force of gravity --- unknown at that time --- is disregarded, the model of de Brahe is entirely equivalent to the Copernican one.

Another notion for which any attempt of locating it within the generic theory of truth leads to similarly extreme consequences is \emph{evental site} . An evental site --- an element ``on the edge of the void'' \cite[p. 175]{BE} --- opens up a situation to the possibility of an event \cite[p. 179]{BE}. Ontologically, it is defined as ``a multiple such that none of its elements are presented in the situation'' \cite[p. 175]{BE}. In other words, it is a set such that neither itself nor any of its subsets are elements of the state of the situation. As the double meaning of this word indicates, the state (\emph{\'{e}tat}) in the context of historical situations
 takes the shape of the State (\emph{\'{E}tat}) \cite[p. 104]{BE}. A paradigmatic example of a historical evental site is proletariat --- ``entirely dispossessed, and absent from the political stage'' \cite[p. 69]{E}.

The existence of an evental site in a situation is a necessary requirement for an event to occur. Badiou is very strict about this point: ``we shall posit once and for all that there are no natural events, nor are there neutral events'' \cite[p. 178]{BE} --- and it should be clarified that situations are divided into natural, neutral, and those that contain an evental site. The very matheme of event --- its formal definition is of no importance here --- is based on the evental site \cite[p. 179]{E}. An event raises the evental site to the surface, makes it represented on the level of the state of the situation. Moreover, even if a novelty has the structure of the generic set but it does not emerge from the void of an evental site, it leads to a \emph{simulacrum of truth} instead of a truth \cite[p. 72]{E}, which is one of the figures of Evil \cite[p. 87]{E}. An example of utterly destructive effects of a simulacrum of truth is the Nazis revolution whose source was the ``plenitude'' of the German people \cite[p. 73]{E}.

However, if one takes the mathematical framework of Badiou's concept of event seriously, it turns out that there is no place for the evental site there ---  it is forbidden by the assumption of transitivity of the ground model $M$. This ingredient plays a fundamental role in forcing, and its removal would ruin the whole construction of the generic extension. As it has been already mentioned, transitivity means that if a set belongs to $M$, then each of its elements also belongs to $M$. However, an evental site is a set none of whose elements belongs to $M$. Therefore, contrary to Badiou's intentions, there cannot exist evental sites in the ground model. Using Badiou's terminology one can say that forcing may only be the theory of the simulacrum of truth.

\section{The mathemes of indiscernible and unnameable}

``Thought is nothing other than the desire to finish with the exorbitant excess of the state'' \cite[p. 282]{BE}. Since Cantor's theorem implies that this excess cannot be removed or reduced to the situation itself, the only way left is to take control of it.  A basic, paradigmatic strategy for achieving this goal is to subject the excess to the power of language. Its essence has been expressed by Leibniz in the form of the principle of indiscernibles: there cannot exist two things whose difference cannot be marked by a describable property \cite[p. 283]{BE}. In this manner, language assumes the role of a ``law of being'' \cite[p. 283]{BE}, postulating identity, if it cannot find a difference. Meanwhile --- according to Badiou --- the generic truth is indiscernible: there is no property expressible in the language of set theory that characterizes elements of the generic set. Truth is beyond the power of knowledge, only the subject can support a procedure of fidelity by deciding what belongs to a truth. This key thesis is established using purely formal means, so it should be regarded as one of the peak moments of the mathematical method employed by Badiou. In order to assess its grounding and possible limitations, one needs to analyze the matheme of indiscernible as closely as possible.

To some surprise to the reader, Badiou composes the indiscernible out of as many as three different mathematical notions. First of all, he decides that it corresponds to the concept of inconstructible \cite[p. 355]{BE}. Later, he writes though that ``a set $\delta$ is discernible (\ldots) if there exists (\ldots) an explicit formula $\lambda(x)$ (\ldots) such that 'belong to $\delta$` and 'have the property expressed by $\lambda(x)$` coincide'' \cite[p. 367]{BE}. Finally, at the outset of the argument designed to demonstrate indiscernibility of truth he brings in yet another definition: ``let us suppose the contrary: the discernibility of $G$. A formula thus exists $\lambda(x,a_1,\ldots, a_n)$ with parameters $a_1 \ldots,a_n$ belonging to $M[G]$ such that for an inhabitant of $M[G]$ it defines the multiple $G$'' \cite[p. 386]{BE}. Summarizing, the discernibility is understood as:

\begin{enumerate}
\item constructibility
\item  definability by a formula $F(y)$ with one free variable and no parameters. In this approach, a set $a$ is definable if there exists a formula $F(y)$ such that $b$ is an element of $a$ if and only if $F(b)$ holds.
\item  definability by a formula $F(y,z_1 \ldots,z_n)$ with parameters. This time, a set $a$ is definable if there exists a formula $F(y,z_1, \ldots,z_n)$ and sets $a_1,\ldots,a_n$ such that after substituting $z_1=a_1,\ldots,z_n=a_n$, an element $b$ belongs to $a$ if and only if $F(b,a_1,\ldots,a_n)$ holds.
\end{enumerate}

Even though in ``Being and Event'', Badiou does not explain the reasons for this variation, it clearly follows from his other writings (such as \cite[p. 135]{C}) that he is simply convinced that these notions are equivalent. It should be emphasized then that this is not true: a set may be discernible in one sense, but indiscernible in another. First of all, the last definition has been included probably by mistake because it is trivial. Every set is discernible in this sense because for every set $a$ the formula $F(y,x)$ defined as '$y$ belongs to $x$` defines $a$ after substituting $x=a$. Accepting this version of indiscernibility would lead to the conclusion that truth is always discernible, while Badiou claims, and, supposedly, demonstrates that it is not. In particular, the proof of indiscernibility of truth presented by Badiou on page 386 of ``Being and Event'', which is based on this definition of indiscernibility, is incorrect\footnote{There are several flaws in the proof but the most important one is that Badiou wrongly assumes that an element defined by a formula with parameters in the ground model must be an element of the ground model as well. This is not true as the following discussion will show.}.

Is it not possible to choose the second option and identify discernibility with definability by a formula with no parameters? After all, this notion is most similar to the original idea of Leibniz --- intuitively, the formula $F(y)$ expresses a property characterizing elements of the set defined by it. Unfortunately, this solution does not warranty indiscernibility of the generic set either. In 2005, Eli Enayat (\cite{Enayat}) published a paper, where he proves the possibility of generic extensions all of whose elements --- including the generic set --- are definable by a formula with no parameters. Therefore, in this approach truth may be seized by knowledge replacing the intervening subject.

As a matter of fact, if we assume that in ontology discernibility corresponds to constructibility, then Badiou is right that the generic set is necessarily indiscernible. However, constructibility is a highly technical notion, and its philosophical interpretation seems very problematic. Let us take a closer look at it.

The class of constructible sets --- usually denoted by the letter $L$ --- forms a hierarchy indexed or `numbered' by ordinal numbers. Without getting into details of the definition of ordinal number --- closely related to that of cardinal number --- the inductive procedure of constructing the constructible hierarchy goes as follows. The lowest level $L_0$ is simply the empty set. Assuming that some level --- let us denote it by $L_\alpha$ --- has been already constructed, the next level $L_{\alpha+1}$ is constructed by choosing all subsets of $L_\alpha$ that can be defined by a formula (possibly with parameters) \emph{bounded} to the lower level $L_\alpha$.

Bounding a formula to $L_\alpha$ means that its parameters must belong to $L_\alpha$ and that its quantifiers are restricted to elements of $L_\alpha$. For instance, the formula 'there exists $z$ such that $z$ is in $y$` simply says that $y$ is not empty. After bounding it to $L_\alpha$ this formula takes the form 'there exists $z$ in $L_\alpha$ such that $z$ is in $y$`, so it says that $y$ is not empty, and some element from $L_\alpha$ witnesses it. Accordingly, the set defined by it consists of precisely those sets in $L_\alpha$ that contain an element from $L_\alpha$.

After constructing an infinite sequence of levels (or --- strictly speaking --- a limit sequence of levels) the level directly above them all is simply the set of all elements constructed so far. For example, the first infinite level $L_\omega$ consists of all elements constructed on levels $L_0, L_1, L_2, \ldots$.

As a result of applying this inductive definition, on each level of the hierarchy all the formulas are used, so two distinct sets may be defined by the same formula. On the other hand, only bounded formulas take part in the construction. The definition of constructibility offers too little (because only bounded formulas are accepted) and too much at the same time (because many sets can be defined by one formula). This technical notion resembles the Leibnizian discernibility only in so far as it refers to formulas. In set theory there are more notions of this type though.

To realize difficulties involved in attempts to philosophically interpret constructibility, one may consider a slight, purely technical, extension of it. Let us also accept sets that can be defined by a formula $F(y,z_1,\ldots,z_n)$ with constructible parameters, that is, parameters coming from $L$. Such a step does not lead further away from the common understanding of Leibniz's principle than constructibility itself: if parameters coming from lower levels of the hierarchy are admissible when constructing a new set, why not admit others as well, especially since this condition has no philosophical justification?

Actually, one can accept parameters coming from an even more restricted class,  e.g., the class of ordinal numbers --- and these are known to be constructible. Then we will obtain the notion of \emph{definability from ordinal numbers} $(OD)$. This minor modification of the concept of constructibility --- a  relaxation of the requirement that the procedure of construction has to be restricted to lower levels of the hierarchy --- results in drastic consequences. Kenneth McAloon (\cite{McAloon}) proved in 1972 that the generic set as well as all other elements of the generic extension may be definable from ordinal numbers. Therefore, replacing constructibility with a concept very similar to it ---  one essentially identical from the point of view of Badiou's philosophical motivations --- leads to an ontology allowing for discernibility of truth.

Another example of such a heterogeneous composition is the matheme of \emph{unnameable}. In ``Being and Event'' this notion functions as a synonym of the indiscernible, however in later writings of Badiou ---  especially ``Conditions'' and ``Ethics'' ---  becomes a distinct, autonomous concept. Referring to the Lacanian psychoanalysis, Badiou says: ``in the field determined by a situation and the generic becoming of its truth, a real is attested to by a term, a point, and only one, at which the power of truth is suspended.  There is only one term in relation to which no anticipatory hypothesis on the generic subset permits us to force a judgment, (\ldots) no naming is appropriate for this term. That is why I call it unnameable'' \cite[p. 141]{C}. Also, the unnameable understood in the same manner --- as a ``point that the truth cannot force'' \cite[p. 85]{E} --- witnesses the ``powerlessness of truth'' in Badiou's ethical thought. A totally powerful truth  ---  its subject language reaching all elements of the situation ---  leads to a ``disaster'', which is another figure of Evil, along with a simulacrum of truth \cite[p. 85]{E}.

Thus in set theory, the unnameable has a precisely defined form: it is the \emph{unique} element of the generic extension such that no condition forces a sentence of the forcing language referring to this element. Unfortunately, the mathematical content of this definition turns out to be empty ---  forcing unequivocally rules out the existence of the unnameable. This is because one of the basic features of the forcing language is (\cite[Th. 14. 7 ii) a)]{Jech}) that if no condition forces some sentence $S$, then every condition forces the negation of $S$. In the generic extension, the unnameable is not.

Yet Badiou's position is that mathematics indicates the possibility of the unnameable. He points to a construction due to Furkhen ``in which one term exists, and only one, which cannot receive a name in the sense that it cannot be identified by a formula in the language'' \cite[p. 143]{C}. A familiar substitution can be recognized here. Initially, the unnameable is defined in terms of the forcing language, however Furkhen's construction refers to definability by a formula. It should be emphasized that this object is not even a model of set theory \cite[p. 120]{C}; it requires a richer  language, and it does not satisfy axioms of set theory. Therefore it cannot serve as a ground model nor a generic extension --- forcing has simply no use in this framework. Moreover, the unnameable becomes the indiscernible again with an additional requirement of uniqueness. Is a truth a real of the situation then? Such a suggestion never appears in Badiou's writings. Otherwise, there always exist at least two indiscernible elements --- a truth and a real of the situation --- even though the unnameable is supposed to be unique. In presence of these overwhelming formal and interpretative difficulties, one should rather conclude that the matheme of unnameable is an ill-conceived concept.

\begin{center}
***
\end{center}

The mathematical form of the undecidable implies that a truth never overrules the fundamental conceptual framework of a situation from before the occurrence of an event. This consequence remains in sharp contrast with basic examples of the effect of event considered by Badiou. An evental site cannot open up a situation to an evental truth because forcing rules out the very existence of an evental site in ground models. As a result, the generic set may possibly describe the structure of the simulacrum of truth --- using Badiou's terminology ---  but not the structure of truth itself. Two two remaining mathemes --- the indiscernible and the unnameable --- are built out of mathematical concepts that do not fit together; moreover, their properties are not as Badiou claims.

This list of problems can be extended.  The ``name of the event'' is not ``prohibited by being'' \cite[p. 184]{BE} because the axiom of foundation may be consistently replaced with its negation --- for example, the Aczel anti-foundation axiom is consistent with the remaining axioms of set theory. Finite sets are not necessarily definable, even though Badiou tries to prove the opposite \cite[p. 137]{C}, mistakenly identifying finite sets with numerals. In the so called nonstandard models of set theory the standard omega is finite and non-definable. Generic sets are not coextensive with unconstructible sets \cite[p. 135]{C} -- for example, the Ground Axiom implies it. It seems though that the presented discussion of the mathemes of indiscernible, undecidable, unnameable, and evental site is sufficient to defend the main thesis of this paper. Mathematics --- forced to accept compromises going definitely too far --- responds with outcomes which are hostile to fundamental philosophical motivations of Badiou's doctrine. Despite some points of convergence, his generic theory of truth and his philosophy of event can coexist only at a price of selective and instrumental interpretation of the mathematical component. Therefore one has to conclude that ``Being and Event'' provides no grounding for a deep ontological structure behind the realms of science, art, love and politics, and that the mathematical formulation of the theory of event has no positive content.

config: arxiv


TITLE: Stirling numbers practice question 1
QUESTION [2 upvotes]: This is a practice question and not HW
Q. Let $A=\{1,2,3,4,5,6,7\}$ and $B=\{v,w,x,y,z\}$. Determine the number of functions $f:A \longrightarrow B$ where
a) $f(A)=\{v,x\}$
Ans. $2!*S(7,2)$
I understand that $S(7,2)$ is the total number of onto function from $A$ to $B$ but why multiply by $2!$? I do see see that $f(x)=(v,w)$ has two items so it certainly has something to do with that but why the factorial and why multiply it. Some intuition would be greatly appreciated.
b) $|f(A)|=2$
here the answer is $(5C2)(2!)S(7,2)$ Again why the $2!$ and why $5C2$

REPLY [3 votes]: $S(n,k)$, or ${n\brace k}$, to use my preferred notation, is the number of ways to divide $n$ labelled objects amongst $k$ unlabelled containers so that no container is empty. If we then attach distinct labels to the $k$ containers — the numbers $1$ through $k$, for instance — we can do it in $k!$ different ways, one for each permutation of the containers. Thus, there are $k!{n\brace k}$ ways to divide $n$ labelled objects amongst $k$ labelled containers in such a way that no container is empty. In the case of a function from an $n$-element set onto a $k$-element set, you’re assigning each of the $n$ individually identifiable elements of the domain to one of the $k$ individually identifiable elements of the range, so the number of functions is $k!{n\brace k}$, not $n\brace k$. In particular, in this problem it’s $2!{7\brace 2}$.
In part (b) you have $2!{7\brace 2}$ functions for each possible $2$-element subset of $B$. Since $|B|=5$, there are $\binom52$ such subsets, each contributing $2!{7\brace 2}$ functions, for a total of $\binom52\cdot2!\cdot{7\brace2}$ functions.


TITLE: What is the universal definition of the order parameter that is valid irrespective of the nature of the phase transition?
QUESTION [2 upvotes]: Plausible definition Consider a phase transition from phase 1 to phase 2. The order parameter is zero in one of the phases 1 or 2 and nonzero in the other.
For example, in normal (phase 1) to superfluid (phase 2) transition, the order parameter is zero in the normal phase and nonzero in the disordered phase. So in this case, the above definition works good.
However, in the case of gas (phase 1) to liquid (phase 2) transition, the order parameter is taken to be $\mathcal{O}=\rho_{liq}-\rho_{gas}$. But $\mathcal{O}$ is nonzero in both the phases 1 and 2, and only vanishes above the critical temperature $T_c$. So, in this case, the above definition doesn't hold good.
Does it mean that the definition

Consider a phase transition from phase 1 to phase 2. The order parameter is zero in one of the phases 1 or 2 and nonzero in the other.

is wrong?
Is there a universal definition of order parameter such that it hold's good in both the cases?

REPLY [1 votes]: The order parameter is discontinuous for first order phase transition at the transition point. However, it need not be zero in any of the phases between which the first order transition takes place. For example, the liquid-to-gas transition or vice-versa below $T_c$, the order parameter is $\rho_{liq}-\rho_{gas}$, is non-zero in both liquid and gas but changes discontinuously. This also holds good for first order magnetic transition from up-aligned Ising ferromagnet to the down-aligned phase.
However, the same order parameter $\rho_{liq}-\rho_{gas}$ is zero in the phase above $T_c$ (the disordered phase) and nonzero below it for the second order phase transition at $T_c$. This also works for paramagnetic to ferromagnetic transition at $T_c$. This resolves the problem.


TITLE: Using Cramer's Rule
QUESTION [0 upvotes]: The three equations given are:
\begin{cases}
x+y-z=4 \\
x-2y+3z=-6 \\
2x+3y+z=7
\end{cases}
I am having trouble putting it into the matrix format, which in turn is hindering my ability to solve for the determinant. I am also not sure how to solve for $x$, $y$, and $z$ with Cramer's Rule.

REPLY [1 votes]: We start by writing the equations like so:$$\begin{pmatrix}
    1 & 1 & -1\\
    1 & -2 & 3\\
    2 & 3 & 1
\end{pmatrix}
\begin{pmatrix}
  x \\ y\\ z
\end{pmatrix} =
\begin{pmatrix}
 4\\ -6\\ 7
\end{pmatrix}.$$
Now, let's call the $3\times 3$ matrix $A$ and the vector on the right hand side of the equation $b$. Then, we first observe $$\det(A)=-13.$$ In order to use Cramer's rule, we replace the relevant column with $b$, calling this matrix $A_i$ for $i\in\{x,y,z\}$ and work out the determinant of this new matrix and divide it by $\det(A)$. So, carrying this procedure out gives $$\det(A_x)=\det\begin{pmatrix}
    4 & 1 & -1\\
    -6 & -2 & 3\\
    7 & 3 & 1
\end{pmatrix}=-13,$$ $$\det(A_y)=\det\begin{pmatrix}
    1 & 4 & -1\\
    1 & -6 & 3\\
    2 & 7 & 1
\end{pmatrix}=-26,$$ $$\det(A_z)=\det\begin{pmatrix}
    1 & 1 & 4\\
    1 & -2 & -6\\
    2 & 3 & 7
\end{pmatrix}=13.$$ This then allows us to conclude $$x=1,\quad y=2,\quad z=-1.$$
However, as Bernard alludes to in the comments, I think you may appreciate at this point that Cramer's rule is quite inefficient and I would suggest using reduced row echelon form to obtain the solution set to these sorts of problems instead.


TITLE: Calculate a closed interval that contains the roots of a polynomial function
QUESTION [0 upvotes]: Calculate a closed interval of $\mathbb{R}$ that contains the roots of
$$ x^5 + x^4 + x^3 + x^2 + 1 $$
Should I use the Intermediate Value Theorem and try to guess multiple points in which I know the function crosses the x-axis? I am pretty sure there is a smarter way of doing this, but I am not quite getting there.

REPLY [0 votes]: Multiply your polynomial with $x-1$ to get $x^6-1$. Now you can see that all roots of your original polynomial are roots of unity. In particular you have for a root $\xi$ of $x^5+\ldots+1$, that $\vert\xi\vert=1$. What will your intervall be?


TITLE: big O inequality
QUESTION [0 upvotes]: In $e^{-\omega/\epsilon}\leq10^{-9}$ why is $\omega=O(\epsilon)$ and in $e^{-\omega/\epsilon}\leq \epsilon$ why is $\omega=O(\epsilon\ln1/\epsilon)$ where $\epsilon$ is a very small parameter.

REPLY [1 votes]: (I'll post my comment above as an answer since it seems to address what the OP was looking for.)
In the reference given for the source of this question it states

The width of the boundary layer [...] could be taken to be the smallest value $\omega$ such that $e^{-\omega/\epsilon} \leq 10^{-9}$, say.  In this case it is clear that $\omega = O(\epsilon)$.

As shown in this related answer, the smallest such $\omega$ is $\omega = (9\log 10)\epsilon$, which is indeed $O(\epsilon)$.
It's not true that all $\omega$ which satisfy $e^{-\omega/\epsilon} \leq 10^{-9}$ are $O(\epsilon)$ (for example take $\omega = (9\log 10)\sqrt{\epsilon}$), but the smallest one, the one you care about, certainly is.  Dealing with the other inequality is similar.


TITLE: Ball A is dropped from the top of a building. At the same instant another bowl B is thrown vertically upward from the ground.
QUESTION [0 upvotes]: When the balls collide, they are moving in opposite direction and the speed of A is twice the speed of B. At what fraction of the height did the collision occur?

Now I have solved this question a bit differently. I did not arrive at the right answer, so I need to know what I have to fix.
Let the building be of height H and let the balls collide at h and time t
Distance travelled by A
$$H-h=\frac 12 gt^2$$
So $$t^2=\frac 2g (H-h)$$
And distance covered by B is
$$h=ut - \frac 12 gt^2$$
Subsisting the value of t
$$h=u\sqrt {\frac{2(H-h)}{g}} - (H-h)$$
Squaring after doing appropriate simplification
$$H^2=\frac{2u^2(H-h)}{g}$$
I don’t know how to proceed further. Please help.

REPLY [1 votes]: The distances traveled by the two balls
$$h =ut - \frac 12 gt^2$$
$$H - h = \frac 12 gt^2$$
Take their ratio
$$ \frac {h}{H-h} = \frac{2u}{gt} -1\tag{1}$$
Let $u_a$ and $u_b$ be the velocities at collision, then $gt$ is related to them via
$$gt = u-u_b = u_a\tag{2} $$
And, at collision,
$$u_a= 2u_b\tag{3}$$
From (2) and (3), we get $u=3u_b$, which leads to
$$gt = u-u_b = \frac23 u$$
Plug above $gt$ into (1) to get
$$ \frac {h}{H-h} = \frac 21$$
Thus, they collide at
$$h=\frac23 H$$.


TITLE: Large cardinal axiom: everything that happen once must happen an unbounded number of times
QUESTION [13 upvotes]: I remember reading something about a large cardinal axiom saying something like

If some cardinal $\kappa$ has some property $P$, then there should be a proper class of cardinals with the property $P$.

Of course this is inconsistent if you allow any property $P$, but this was used to justify for example the existence of uncountable inaccessible cardinals, because $\aleph_0$ is inaccessible, so there should be a proper class of inaccessible cardinals (because having only one inaccessible cardinal is not very homogeneous, and the ordinal hierarchy should be homogeneous).
Unfortunately I don't remember where I read that.
My questions are: Is there a precise formalisation of this axiom, and (if there is one) how high is it in the large cardinal hierarchy?

REPLY [14 votes]: Here are two contexts in which such conclusions follow.
Strong reflection axioms. Consider the strong reflection axiom, sometimes denoted $V_\delta\prec V$, which is axiomatized in the language of set theory augmented with a constant symbol for $\delta$, axiomatized by the assertions $$\forall x\in V_\delta\ \ [\varphi(x)\iff \varphi^{V_\delta}(x)].$$
This theory is a conservative extension of ZFC, since every finite subset of the theory can be interpreted in any given model of ZFC by the reflection theorem. Meanwhile, in this theory, if the cardinal $\delta$ (or any larger cardinal) has a property $P$ expressible in the language of set theory, then $V$ satisfies $\exists\kappa P(\kappa)$, and so $V_\delta$ also satisfies this assertion, so there is a $\kappa\lt\delta$ with $P(\kappa)$. Similarly, if the collection of cardinals $\kappa$ with $P(\kappa)$ is bounded, then $V_\delta$ would have to agree on the bound by elementarity, but it cannot agree on the bound since $\delta$ itself has the property. So the collection of cardinals with property $P$ must be unbounded in the ordinals.
A stronger formulation of the strong reflection axiom includes the assertion that $\delta$ itself is inaccessible (or more), in which case it carries some large cardinal strength. It is exactly equiconsistent with the assertion "ORD is Mahlo", meaning the scheme asserting that every definable stationary proper class contains a regular cardinal, which is weaker in consistency strength than a single Mahlo cardinal.
Another seemingly stronger formulation of the theory, but still conservative over ZFC, is due to Feferman, and this asserts that there is a closed unbounded proper class $C$ of cardinals $\delta$, all with $V_\delta\prec V$. This theory can be stated as a scheme in the language of set theory augmented by a predicate symbol for $C$. Feferman proposed it as a suitable substitute and improvement of the use of universes in category theory, because it provides a graded hierarchy of universe concepts, which moreover agree between them and with the full universe on first-order set-theoretic truth.
The maximality principle. The maximality principle is the scheme asserting that any statement $\varphi$ which is forceable in such a way that it continues to be true in all further forcing extensions, is already true. This axiom is the main focus of my article, "A simple maximality principle", JSL 62, 2003, and was introduced independently by Stavi and Vaananen. The axiom asserts in short, that anything that you could make permanently true in forcing extensions, is already permanently true. The point now is that under MP, one gets your phenomenon:
Theorem. Under MP, if there is any inaccessible cardinal, then there is a proper class of such cardinals. And the same for Mahlo cardinals and many other large cardinal concepts.
Proof. Assume MP. If there are no inaccessible cardinals above some ordinal $\theta$, then consider the forcing to collapse $\theta$ to be countable. In the resulting forcing extension $V[G]$, there will be no inaccessible cardinals at all, and there never will be such cardinals in any further extension (because any inaccessible cardinal of $V[G][H]$ would also be inaccessible in $V[G]$). Thus, in $V$ the assertion that there are no inaccessible cardinals is forceably necessary, and so by MP, it must already be true. Thus, under MP, either there are no inaccessible cardinals or there are a proper class of them. The same argument works with Mahlo cardinals or any large cardinal concept that is downwards absolute. QED


TITLE: A question of finite Group
QUESTION [2 upvotes]: Let $G = A_5$, the alternating group of degree five,. Let $\pi = \{2,3\}.$
Prove that $M$ is a maximal $\pi$-subgroup of $G$ if, and only if, $M\cong A_4$ or $M\cong S_3$, where $A_4$ is the alternating group of degree four and $S_3$ is the symmetric group $3$.

REPLY [0 votes]: Clearly we have both subgroups $\,S_3\,,\,A_4\;$ embedded in $\,A_5\,$ , and since a $\,\pi$-subgroup here would have to have order$\,\in\{2,3,4,6,12\}\;$ , we at once get that $\,M\cong A_4\,$ is a maximal $\,\pi$-subgroup , and since $\,S_3\rlap{\;\;/}\le A_4\,$ then this one's also a maximal $\;\pi$-subgroup .


TITLE: Estimating grader bias/variance and MLE test scores given multiple graders assigned to grade each test
QUESTION [2 upvotes]: Suppose we have $m$ graders and $n$ students, and we want to grade a test so that $k$ graders are assigned to grade to each test, and all graders grade the same number of tests. (I realize $m,n,k$ have to satisfy certain properties to make this "perfect assignment" possible, but I'd rather just skip this point and assume it's true). Also, to make things interesting, let's assume the assignment of graders to tests is random (so it's not like a group of graders all have the same set of tests to grade).
Furthermore, let's assume that each grader $i$ has a mean bias $\mu_i$ and a variance $\sigma_i^2$ for that bias associated with their grading, and the bias they apply to each test they grade is sampled independently from a normal distribution with these parameters. And each test $j$ has a "true grade" $c_j$. So then if grader $i$ is assigned to grade test $j$, then the grade they assign will be $c_j + x_{ij}$ where $x_{ij}$ is the sampled bias from the normal distribution with parameters $\mu_i$ and $\sigma_i^2$.
If the $\mu_i$ and $\sigma_i^2$ are unknown, how do we find the maximum likelihood values for the true grade scores $c_j$? If using a prior for graders' parameters is required I guess I'm ok with that. I would also like to know the MLE (or MAP if we go Bayesian) values for the grader parameters $\mu_i$ and $\sigma_i^2$. The idea being that graders with lower estimated variance should be preferred to those with higher variance, if we want as accurate of assigned grades as possible in the future.
I've phrased this in terms of test grading for clairty, but it's actually for an "active learning" problem in machine learning that we are very interested in in our lab, hence insights on this problem could really help.

REPLY [1 votes]: Ok, here's what I have so far:
For the MLE:
You're right. The MLE formula will not have a closed form solution. But, the outline of the formula seems simple enough. Treating each test score as an observation, we have $T_{ij} = c_i + x_{ij}$ ($i$ refers to students and $j$ markers)
Now lets make some assumptions. Suppose that the "true" test scores are also normally distributed with mean $y$ and variance $\Sigma$. Now, each test score is the sum of independent normally distributed variables. Thus, we have that: $TS_{ij}$~$N(y+\mu_j,\Sigma+\sigma_j^2)$
If we knew the values of the types, then the likelihood would be easily given by $L(TS_{ij} | \mu_j,\sigma_j) = \frac{1}{\sqrt{\Sigma+\sigma_j^2}}\phi(\frac{TS_{ij}-y-\mu_j}{\sqrt{\Sigma+\sigma_j^2}})$ where $\phi$ is the standard normal pdf. But, since we do not know each persons type, they need to be integrated out of the likelihood.
$L_{ij}=\int \frac{1}{\sqrt{\Sigma+\sigma_j^2}}\phi(\frac{TS_{ij}-y-\mu_j}{\sqrt{\Sigma+\sigma_j^2}})dF(\mu_j,\sigma_j)$
In terms of identification. I focus on the means only.
Since each grader is assigned tests at random, then on average the average grader's test score will be $y + \mu_j$ -- the average child's test score plus the grader's average bias. There will be $M$ such equations. Similarly, the average of the same test $j$ for the $k$ graders will be $y_j + E[\mu_i]$ -- the child's average test score plus the average marker bias. There are $N$ such equations here. (Note: For simplicity I've imposed the Law the Large numbers here)
This suggests you might have an identification problem. You have $M+N$ equations with $M+N+2$ unknowns. This would seem to suggest some normalizations will be required. For example, you can normalize the average test score and bias to be $0$. $y=E[u_j]=0$


TITLE: Prove that $\mathcal L(V,W)$ is isomorphic to $\mathbb F^{m,n}$
QUESTION [0 upvotes]: I'm working through the proofs in Axler's Linear Algebra Done Right but I can't quite crack this one. It's the proposition 3.60, the part that I can't quite crack is the surjectivity proof. Axler says that "to prove that $\mathcal M$ is surjective, suppose $A\in \mathbb F^{m,n}$. Let T be the linear map from $V$ to $W$ such that
$$T_{v_k}=\sum_{j=1}^mA_{j,k}w_j$$
for $k=1,...,n$. Obviously $\mathcal M(T)$ equals A, and thus the range of $\mathcal M$ equals $\mathbb F^{m,n}$, as desired."
The whole thing transcends me, if anyone knows of an alternate proof or can give me a hint on this I'd appreciate it.

REPLY [0 votes]: You said that $\mathcal{M}(T)=A$. What do you have to show to establish surjectivity? Every $A$ corresponds to a $T$, which is the case.


TITLE: Verify the limit using the definition of convergence of a sequence
QUESTION [1 upvotes]: I know that the definition is:
"A sequence $(s_n)$ is said to converge to the real number $s$ provided that for every $\epsilon > 0$ there exists a natural number $N$ such that for all $n \in \mathbb{N}$, $N \leq n$ implies that $|s_n - s | < \epsilon$".
So here is an exercise in the problem section of the chapter:
"Verify the limit using the definition of convergence of a sequence that:
$\lim_{n \to \infty} \frac{\sqrt[]{n}}{n+2} = 0$"
I don't know how to really start. I guess, given any $\epsilon$ we want to make $|\frac{\sqrt[]{n}}{n+2} - 0| < \epsilon$. Looking at:The definition of a sequence converging.
So, lets pick a value for epsilon, $\epsilon = 0.5$. Now we have to find a natural number $N$ s.t. for $n \in \mathbb{N}$, $|s_n - s| < \epsilon$. So, we have to solve for $n$ in $\frac{\sqrt[]{n}}{n+2} < \frac{1}{2}$.
Couldn't that be quite messy with other limit problems? I suppose we could use the fact if $a < b$ and $b < c$, then $a < c$, but i'm not sure how exactly to find something larger than $\frac{\sqrt[]{n}}{n+2}$ .

REPLY [0 votes]: We want to prove that $\left | \dfrac{\sqrt{n}}{n+2} \right | <\epsilon$ for a sufficiently large $n$. Since $\dfrac{\sqrt{n}}{n+2} > 0$:
$$\dfrac{\sqrt{n}}{n+2} <\epsilon$$
$$\sqrt{n}< \epsilon (n+2)$$
Since both members of the inequality are positive, we can square them and the inequalilty holds:
$$n < \epsilon^2 (n+2)^2$$
$$n - \epsilon^2 n^2 - 4\epsilon^2 n - 4\epsilon ^2 < 0$$
$$-\epsilon^2 n^2 + (1-4\epsilon^2)n - 4\epsilon^2 < 0$$
Which is true for a sufficiently large $n$.


TITLE: What are the odds that 1 suit shows up at least 3 times in the first 8 cards chosen?
QUESTION [0 upvotes]: (We keep inventing betting games...and I'm curious if I'm giving the right odds :)
In a normal, shuffled, deck of cards,
What are the odds that [Lets say, Hearts] shows up at least 3 times in the first 8 cards chosen?
Cards are not returned to the deck between each pick.

REPLY [3 votes]: Let's warm up with the easier question of getting exactly three hearts.
The number of ways to choose 8 cards from a deck of 52 is the
binomial coefficient $52\choose 8$. The number of ways of
getting exactly three hearts is ${13\choose 3}\times{39\choose 5}$.
Here "13" is the number of hearts (of which we take 3) and "39" is
the number of non-hearts (of which we take 5).
The probability of getting exactly three hearts is, therefore,  the ratio
$${{13\choose 3}\times{39\choose 5} \over {52\choose 8}}.$$
For the probability of getting 3 or more hearts, you repeat
this pattern replacing 3 with 4,5,6,7 and 8, then add the results.
The probability you want is
$${{13\choose 3}\times{39\choose 5} \over {52\choose 8}}+{{13\choose 4}\times{39\choose 4} \over {52\choose 8}}
+{{13\choose 5}\times{39\choose 3} \over {52\choose 8}}+{{13\choose 6}\times{39\choose 2} \over {52\choose 8}}
+{{13\choose 7}\times{39\choose 1} \over {52\choose 8}}+{{13\choose 8}\times{39\choose 0} \over {52\choose 8}}.$$
When you do the calculations, this turns out to be $ 118943/378350 = 0.31437$.
As ncmathsadist points out, for calculation purposes, it is easier to consider the opposite
problem of getting two or fewer hearts, and subtracting from 1. The probability you want can
also be written
$$1- {{13\choose 0}\times{39\choose 8} \over {52\choose 8}}-{{13\choose 1}\times{39\choose 7} \over {52\choose 8}}
- {{13\choose 2}\times{39\choose 6} \over {52\choose 8}} .$$
Problems of this kind, where you sample with replacement, give hypergeometric probabilities.
If you had replaced the card after each draw, you would use binomial probabilities instead.
http://en.wikipedia.org/wiki/Binomial_probability_distribution
http://en.wikipedia.org/wiki/Hypergeometric_distribution


TITLE: Fourier Transform of a Horizontally Scaled Function
QUESTION [1 upvotes]: For $f(x)$ we have that
$$
\widehat{f}(\xi) = \int_\mathbb{R} e^{-2 \pi i x \xi} f(x)
$$
Now let $s \in \mathbb{R}$ and consider
$$
\widehat{f(sx)}(\xi) = \int_\mathbb{R} e^{-2 \pi i (sx) \xi} f(sx) = \int_\mathbb{R} e^{-2 \pi i (x) \xi} f(x)*{d \over ds}(1/s) = -{1 \over s^2}\int_\mathbb{R} e^{-2 \pi i (x) \xi} f(x) = -{1 \over s^2} \widehat{f}(\xi)
$$
through a change in variable $x \mapsto (1/s)x$.
Question: Is this calculation correct?

REPLY [2 votes]: No. Let $\varphi(x)=sx$, so the question is addressing the relationship between $\widehat{f}$ and $\widehat{f\circ\varphi}$. We have
\begin{align*}
\widehat{f\circ\varphi}(\xi)&=\int f\circ\varphi(x)e^{-2\pi ix\xi}dx\\
&=\int f(sx)e^{-2\pi ix\xi}dx\\
&=\int f(sx)e^{-2\pi i(sx)(\xi/s)}\dfrac{d(sx)}{s}\\
&=\dfrac{\text{sgn}(s)}{s}\int f(x)e^{-2\pi ix\xi/s}dx\\
&=\dfrac{\text{sgn}(s)}{s}\widehat{f}(\xi/s).
\end{align*}


TITLE: Triple integral of a vector field
QUESTION [0 upvotes]: Let V be the region in $\mathbb{R}^3$ defined by the inequality $x^2 +y^2 +z^2 ≤ 1$, let $S$ be the surface bounding $V$ and $\vec{v}$ be the vector field $\vec{v} = r^3\vec{r}$. Calculate$\displaystyle \iiint_{V} \nabla \cdot \vec{v} \,  dx \, dy \, dz$ by any method.

I've difficulty understanding the given vector field. What's $r$ and what's $\vec{r}$?

REPLY [1 votes]: The region $x^2 + y^2 + z^2 \leq 1$ is the sphere of radius $1$ centered about the origin.  Given a point $\langle x, y, z \rangle$:
$$
\vec{r} = \langle x, y, z \rangle \\
|\vec{r}| = \sqrt{x^2 + y^2 + z^2}
$$
Thus, using the Divergence Theorem:
$$
\int_V \nabla \circ\vec{v} dV = \oint_{\partial V} \vec{v}\circ \hat{n}dA
$$
But since the boundary is the surface of a sphere--and $\vec{v}$ always points out radially, $\vec{v}$ is always perpendicular to the boundary surface.  $\vec{v}\circ\hat{n} = |\vec{v}|$ for all points on the boundary of the sphere.


TITLE: Lebesgue measurability of a set
QUESTION [6 upvotes]: Prompted by this question I was looking for $A \subset (0,1)$ such that for any interval $(a,b)\subset (0,1), A \cap (a,b)$ and $A^c \cap (a,b)$ are both uncountable.  One such $A$ is the set of all numbers that have a finite number of $1$'s in their base $3$ expansion.  As no choice was used in the construction, it should be Lebesgue measurable, but I can't prove it.  How is it proved?

REPLY [5 votes]: For each $n$, let $A_n$ be the subset of elements of $A$ that have at least a $1$ in the first $n$ digits of their ternary expansion, but no $1$s after the $n$th digit. Your set $A$ is equal to the union of the $A_n$s and the Cantor set. Each $A_n$ is a finite union of scaled translates of the Cantor set (take $3^{-n}C$, where $C$ is the Cantor set, to get all numbers that have ternary expansion with no $1$s and that have $0$ in the first $n$ positions; then you can "translate" by adding a suitable number that has a tail of $0$s and appropriate $1$s in the appropriate coordinates).
So each $A_n$ is a finite union of Lebesgue-measurable sets (scaled Cantor sets are Lebesgue-measurable, and translates of a Lebesgue-measurable set are Lebesgue-measurable), hence Lebesgue measurable. $A$ is a countable union of Lebesgue measurable sets, hence Lebesgue measurable.
Added. As Andres Caicedo points out in the comments below, the argument above shows that the set $A$ is in fact not merely Lebesgue measurable, but Borel.

REPLY [3 votes]: The number $x_n$ in the $n^\text{th}$ place after the radix point in the ternary expansion of $x$ is a measurable function of $x$.  This is true because the floor function is Borel measurable, and $x_n=\left\lfloor 3\cdot\left(3^{n-1}x-\lfloor3^{n-1}x\rfloor\right)\right\rfloor$.  The set $\{x:x_n\neq 1\}$ is measurable for each $n$, and so therefore is the set $$\bigcup_{n=1}^\infty\bigcap_{k=n}^\infty \{x:x_k\neq 1\}.$$

Certain null sets of the second category give further examples of this.  The complement of such a set can be contructed by taking a countable union of closed sets with empty interior whose complements have progressively smaller measure (e.g., using fat Cantor sets).


TITLE: How to convert a dot product of two vectors to the angle between the vectors.
QUESTION [8 upvotes]: I am currently learning to use normalized vectors in the computer games I'm creating.
I've learned that in order to know "the angle" between two vectors, I need to use Dot Product. This gives me a value between $1$ and $-1$. $1$ means they're parallel to each other, facing same direction (aka the angle between them is $0^\circ$). $-1$ means they're parallel and facing opposite directions ($180^\circ$). And $0$ means the angle between them is $90^\circ$.
But I want to know, how to convert the dot product of two vectors, to an actual angle in degrees.
For example, if the dot product of two vectors is $0.28$. How can I convert it to an actual angle, between $0^\circ$ to $360^\circ$?
Thank you

REPLY [11 votes]: Because you want an answer in the range [0,2pi) and not [0,pi], I think that the question's title is misleading.  I think that you are asking, "How can I calculate the angle that will change the direction from vector a to vector b?"
I would not use the arccos formula for dot products, but instead use the arctan2 function for both vectors and subtract the angles.  The arctan2 function is given both x and y of the vector so that it can give an angle in the full range [0,2pi) and not just [-pi,pi] which is typical for arctan.
The angle you are looing for would be given by:
arctan2(b_y, b_x) - arctan2(a_y, a_x)

The result may be a negative angle, but at least it will go from vector a to vector b.  If you want only positive angles, then add 2pi when the angle is negative.
Beware that arctan2 takes y then x.  This is to be similar to arctan which takes y/x as input.


TITLE: Constructing a functional equation that has given solution set.
QUESTION [2 upvotes]: Motivation was that all the functional equations that appear in Olympiad have mainly one solution. for example,
$xf(x^2)f(f(y))+f(yf(x))=f(xy)(f(f(x^2))+f(f(y^2)))$
or
$f(x+y)^2=2f(x)f(y)+max(f(x^2+y^2),f(x^2)+f(y^2))$
You can find infinitely many of them in AOPS site.
Let's consider following functional equations.
$f(xy)=f(x)f(y)$
this clearly has infinitely many non-trivial solutions, that is, $f(x)=x^a$
My question is, does my following conjecture hold?
Conjecture: we call functional equation GOOD if $f(x)=x$ is a solution, and it is consist of algebraic operation of $f$. If there are finitely many non-constant solutions. Then the equation's only non-trivial solution is $f(x)=x$.
My question could be unclear, so basically I'm asking the following.
Can you make a functional equation that only $f(x)=x, f(x)=x^2$ is solution?
More generally, is it possible to construct a functional equation that the solution set is given set?

REPLY [2 votes]: Your conjecture is not correct. For an example possibly more interesting than the trivial cases, like the one suggested by @RobertIsrael, you can take the following functional equation:
$$ g \big( x g ( y ) - x y \big) + x y \big( 2 - y + x y ^ 2 \big) = ( 1 - y ) g ( x y ) + x \big( 1 + x y ^ 2 \big) g ( y ) $$
The only solutions of this equation are given by $ g ( x ) = x $ and $ g ( x ) = 2 x - x ^ 2 $. To show this, define $ f ( x ) = g ( x ) - x $ and you'll have:
$$ f \big( x f ( y ) \big) = ( 1 - y ) f ( x y ) + x ^ 2 y ^ 2 f ( y ) $$
You can see here for a proof showing that the only solutions to this equation are
$ f ( x ) = 0 $ and $ f ( x ) = x - x ^ 2 $.


TITLE: Where am I making a mistake in my evaluation of $A=\frac{1}{\sqrt{2\pi}}\int_{1.96}^{\infty}e^{\frac{-x^2}{2}}dx$?
QUESTION [0 upvotes]: $$A=\frac{1}{\sqrt{2\pi}}\int_{1.96}^{\infty}e^{\frac{-x^2}{2}}dx$$
The following webpage gives this expression and states that ''after integration this yields a value of 0.024998''.
https://statcalc.weebly.com/z-scores-and-probability-distribution.html
I can't figure out how this value was obtained.
I tried squaring the expression and using a double integral but I must be using the wrong limits as I'm not getting the same value.
$$A^2=\frac{1}{2\pi}\int_{1.96}^{\infty}\int_0^{2\pi}e^{\frac{-r^2}{2}}r\ dr\ d\theta$$
$$A^2=\frac{1}{2\pi}\int_{\frac{1.96^2}{2}}^{\infty}\int_0^{2\pi}e^{-u}\ du\ d\theta$$
$$A^2=e^{-1.9208}$$
$$A=0.3827$$
Did I make a mistake along the way or am I using the wrong limits?
I can add more of my steps if necessary.

REPLY [2 votes]: We can make use of $\operatorname{erf}$ -
$$\int \exp\left(\frac{-x^2}{2}\right)\mathrm{d}x=\sqrt{\frac{\pi}{2}}\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)+C$$
So we want to compute
$$A=\frac{1}{\sqrt{2\pi}}\sqrt{\frac{\pi}{2}}\bigg[\lim_{x\to\infty }\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)-\operatorname{erf}\left(\frac{1.96}{\sqrt{2}}\right)\bigg]$$
$$A=\frac{1}{2}(1-0.9500042097...)\approx 0.0249978951...$$
However, as stated by others, $\operatorname{erf}$ is non-elementary.