Questions tagged [foundations]

This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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Why do Mahlo cardinals imply the totality of the finite promise games

Harvey Friedman described his "Finite Promise Games" here and in my opinion a more clear version is described on the Googology Wiki page. He claims that several of the theorems proposed on ...
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Elementarily equivalent models of arithmetic that are not isomorphic.

The book is Predicate Calculus by Goldrei. Given the hint and other similar exercises, this is the only way I know how to go about this: Take the set $\text{Th}(\mathcal{N}) \cup \{ \textbf{c} \not = \...
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What is meant by a "Cantorian sense of a graph"?

I didn't get very far in this before I encountered this: $27 × 37 = 999$, then the comment This equality makes sense in the mainstream of mathematics by saying that the two sides denote the same ...
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Kolmogorov's construction of real numbers cardinality of functions that represent real numbers

Hi i am reading about lesser know construction of real numbers by Kolmogorov. In his construction real numbers are defined as a set $\Phi$ of functions $\alpha: \mathbb{N} \rightarrow \mathbb{N}$ that ...
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Domain of identity function as the largest "collections" of all mathematical objects

Consider in a usual functional programming language, like Haskell, that supports parametric polymorphism. Usually, we can define a function as follows: ...
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5 votes
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Why is ZFC incapable of interpreting second-order logic?

Why is ZFC incapable of interpreting second-order logic? Also, when we say this, are we talking about ZFC as a background theory or using ZFC in some different way? I am interested in this answer ...
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Axiomizing meets/joins of a lattice directly instead of quantifiers

The usual strategy of axiomizing logic axiomizes quantifiers first and then defines joins and meets in terms of them later. Can you reverse the order of definitions? I know various logics can be ...
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How to justify the necessity of the Axioms?

I am studying the logical fundaments of mathematics, but very often I have trouble to understand Peano's and ZFC/ZFC axioms. In Tao's book Analysis I, I found very helpful when he points out what ...
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Formal System Where Every Continuous Function is Almost Everywhere Differentiable

Before Weierstrass defined his function, it was believed that every continuous function, $\mathbb{R} \to \mathbb{R}$, is almost everywhere differentiable. Or at least, something approximating that ...
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Can (should?) the Yoneda embedding be formulated in terms of $0$-categories?

Assuming the Grothendieck axiom of universes (see also here or here), let $U_0$ denote the universe of "ZFC sets", i.e. of sets that can be constructed in ZFC alone without assuming axioms ...
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Is the category of sets a model of category theory?

I think usually a model of a theory is a translation to a set ( for example, models of group theory are structured sets like $(\mathbb{R}, +)$, $(\mathbb{Z}, +)$,… etc.). But a collection of all sets, ...
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Is ZFC arithmetically sound?

I apologize that this question is fairly philosophical and not purely mathematical. For the purposes of this question, I would like to take the point of view that that natural numbers are "real&...
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Consistency in Hilbert's Foundations of Geometry

After reading the entry in the Stanford Encyclopedia of Philosophy on Hilbert and Frege’s correspondence regarding the former’s Foundations of Geometry, I am quite puzzled by a claim that is made by ...
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7 votes
1 answer
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Why does my topology textbook (Munkres) define positive integers as the intersection of all inductive subsets of the reals?

This is how the topology textbook I'm reading (Munkres) defines integers: A subset of the real numbers is "inductive" if it contains 1 and $1+x$ for all $x$ in the subset. The intersection ...
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Is there a constructive presentation of the Henstock-Kurzweil integral?

Treating the Riemann integral in a constructive setting is easy and straightforward. Treating the closely related but much more powerful Henstock-Kurzweil integral constructively is almost easy, ...
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Why is the collection of all finite sets of natural numbers not sigma-ideal on N?

I am having a hard time understanding why it won't be sigma ideal. It is an ideal, that means closure under finite unions holds. Then it should be trivial to see that closure under countable unions ...
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1 answer
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Foundation of mathematics and functions

I am interested in the foundations of mathematics. I know some of the ideas of set theory, especially ZFC. One way to define a function $f$ in such theory is to take $E$ and $F$ two sets, $G \in\...
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Mathematicians endorsing platonism -- examples?

Take platonism to be the view that there are abstract mathematical objects which exist independently of us as mathematicians and our language, thought, and practices. Looking at the Stanford ...
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Proving $n\not\in n$ for $n\in\mathbb N$ without regularity

Is there a way to prove in ZF theory without regularity axiom $\forall n\in\mathbb N$, $n\not\in n$? At this point, I haven't proved yet that $\mathbb N$ is a well-ordered set, so a proof of the well-...
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5 votes
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Can we prove "induction on wffs" in a non-circular way?

I've always had the question of whether formal logic, number theory, or set theory "comes first" in foundations, and I've seen questions asking this. However, I recently came to what I think ...
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What mathematical branch doesn't use the definition of function?

Every branch of mathematics I've seen so far works with sets and functions as core building blocks for its definitions. What's a branch in math that use functions very little or maybe not at all? ...
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Is classical projective geometry, or a theory of the real projective plane, complete?

Essentially the title. Tarski's Euclidean geometry is complete. Is the same true of a theory of the real projective plane (as an example of a model of the theory I am interested in: take the extended ...
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4 votes
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Is set theory still interesting without replacement?

The axiom scheme of replacement is very natural, while also avoidable in most "mainstream" mathematical practices; I know there are execptions such as Borel determinacy. My question is ...
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Most presentations of ZF seem to be incomplete

A minor technical issue with ZF (and other set theories such as Morse-Kelley) is that if one isn't careful, the axioms will admit degenerate model, in which there are no sets at all. The axiom of ...
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What's wrong with defining functions as rules?

From Mathematical Proofs: A Transition to Advanced Mathematics — Later in the 19th century, the German mathematician Richard Dedekind wrote: A function φ on a set S is a law according to which to ...
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Why this works for proving ZF axioms system is free of Russell's paradox?

In a book on mathematical logic, the author explains why ZF axioms avoid universal set like this: We may also now show that no universal set exists. Suppose $u$ is a set containing all sets. By ...
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Prove that the set of equivalence classes generated by ~ is uncountable

I am very unsure on how to do this question though I have attempted it. I'm not sure that the statement "if there exists a bijection between a set A and an uncountable set B, then A is ...
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1 vote
2 answers
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Prove that, if n sets are countably infinite, then the Cartesian product of all the sets is countably infinite.

I am not sure if my proof here is sound, please could I have some opinions on it? If you disagree, I would appreciate some advice on how to fix my proof. Thanks $X_1, X_2, ..., X_n$ are countably ...
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1 vote
1 answer
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Q. Is there a formal definition of the word "rigorous"?

I would like to ask if there is a formal definition of the word "rigorous". I think I have a grasp of the concept of "formal stuffs" but I feel like the word "rigorous" ...
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4 answers
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Why not we avoid the phrase "if we assume AC " and take it as granted?

This is slightly different question. First I need to mention that I am neither a mathematician nor a researcher. As an ordinary student the separation " with and without Axiom of choice " ...
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Definition of $x^a$

How is $x^a$ for $x \in \mathbb{R},a>0$ defined? For $x>0$ I think one can define it as $e^{a ln(x)}$. For $x<0$ I am not sure anymore. I think that $x^\frac{3}{5}=(x^5)^\frac{1}{3}$ makes ...
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Histories in Scott-Potter set theory

I have been reading Michael Potter's Set Theory and its Philosophy. I am confused by the concept of a history, which I understand is somewhat unconventional. The definition given is as follows: The ...
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How does one begin to build mathematical logic?

I'm reading a book on mathematical logic by Ebbinghaus, Flum and Thomas. It turns out that set theory is used to build formal languages. I mean, one begins with the definition of an alphabet, which is ...
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Constructing a sequence up to infinity

Consider the sets $x_1, x_2,x_3,\ldots$ I was wondering if we could construct a sequence of sets as follows: \begin{equation} x_1=\{x_2\},\;x_2=\{x_3\},\;\ldots \end{equation} and continue to infinity....
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Prove that for every $X$, $a\in X$, $f:X\to X$ exists a function $u:N\to X$ and $u(0) = a,u(n^+) = f(u(n))$

I am searching for a proof of: For every $X$, $a\in X$, $f:X\to X$ exists a function $u:N\to X$ and $u(0) = a,u(n^+) = f(u(n))$. I found a proof on this site. What is, formally, to define a function ...
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Does separation + reflection prove ZF - extensionality - foundation?

Assume a first-order context. Let $\mathrm{tran}(x)$ denote that $x$ is transitive: $$\mathrm{tran}(x) \leftrightarrow (\forall y \in x) (\forall z \in y) (z \in x)$$ Let $\mathrm{suptran}(x)$ denote ...
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Tao Real analysis: question on function definition (should it be an axiom?)

In Terence Tao's book "Analysis 1", in definition 3.3.1 (function definition), he says Let $X, Y$ be sets, and let $P(x, y)$ be a property pertaining to an object $x \in X$ and an object $y ...
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Is it consistent with $\sf ZF-Fnd$ that no class whose union is the whole universe can be well-ordered?

If $R$ is a binary relation; $\phi $ is a unary predicate; then: $ \neg [R \text { well orders } \phi \land \forall y \, \exists x: \phi(x) \land y \in x ]$ Where: $R \text { well orders } \phi \...
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Heyting's Definition of Spreads and Species

I was reading "Intuitionism - an Introduction, A. Heyting" but I could not understand his definitions of "species" and "spreads" (and also there is a thing which he calls ...
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4 votes
1 answer
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What is the difference between "Peano arithmetic," "second-order arithmetic," and "second-order Peano arithmetic?"

I think this needs to be clarified, so it would be helpful to see an answer to this somewhere. I've seen the following terms: Peano arithmetic. Second-order arithmetic. Second-order Peano arithmetic. ...
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Is there a mereological system treating both parts and wholes?

In the papers I read, the parthood relation is treated as a relation between two parts of one whole. I did not read any paper about the relation between a part of a whole and the whole. How this ...
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3 votes
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Doubts on sets in logic

I am a university student and this is the first year I stumbled upon logic. During our lectures we usually define sets (like the set of formulae, the set of proofs...) and prove theorems on them, ...
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3 votes
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Model Theory in Van Dalen

I am reading Van Dalen's Logic and Structure (5th ed.), and I am confused about the following part (Lemma 4.3.8). $\mathfrak A$ is isomorphically embedded in $\mathfrak B$ $\iff$ $\mathfrak {\hat B}$ ...
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Are the naturals really a subset of the real numbers? [duplicate]

Ok, so this question seems obvious, right? But what I mean is the numbers in the way they are logically / axiomaticaly defined in the foundations of Mathematics. As far as I know, the naturals are in ...
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1 vote
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What is the "Hilbert-Bernays-Gödel axiomatic system"?

In the preface of Fundamentals of Abstract Analysis by Andrew M. Gleason, he writes, The whole book is based on naive set theory... On the other hand, I believe that every theorem and proof will ...
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4 votes
1 answer
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Are there infinitely many proofs of every true mathematical statement? [closed]

If we assume there is no limit to the amount of mathematics we create/discover, is it possible that there are infinitely many proofs about any true mathematical statement? For example, we have Wiles' ...
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Peano axioms set theory

Arithmetic is the core of the study of numbers; from natural numbers you can construct the other numbers. Now, I have heard that set theory is the most fundamental core of mathematics; for this reason ...
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1 answer
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How can I prove the following list is a list of all the injective maps $f:\{1,2,3\}\to\{1,2,3,4\}$ rigorously? (Munkres "Topology 2nd Edition")

I am reading "Topology 2nd Edition" by James R. Munkres. The following exercise is in this book: Exercise 1(a) on p.44 in section 6: Make a list of all the injective maps $$f:\{1,2,3\}\to\{...
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3 votes
3 answers
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Importance of completeness axiom in real analysis

We use Zermelo-Fraenkel-Choice (ZFC) axioms to define natural numbers, then integers and then rational numbers. Then, we define real numbers as limit points of rational Cauchy sequences. I have two ...
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What is the reason for the existence of real numbers? Is it an artifact/side effect of our thought process?

The observation of lengths that can not be represented by rational numbers was noticed if I recall correctly by some Pythagorean disciple over applying the Pythagorean theorem on a triangle with side ...
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