# 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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### A semiring with deductive properties that can replace Aristotelian logic?

There is class of multi-logics, where one unique value is true and several different values are equivalent with false but not with true. $1, F_1, F_2, F_3,\cdots,F_n$, where $F_i \iff F_j$ Define the ...
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### Bourbaki's definition of function

I saw this definition and I got confused by it: "Let E and F be two sets, which may or may not be distinct. A relation between a variable element x of E d a variable element y of F is called a ...
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### Introduction to Metamathematics by S C Kleen. Help with exercise *135b needed.

I'm having trouble with exercise *135b in Introduction to Metamathematics by S. C. Kleene. The ask is to show that: $\vdash 0<a^{'}$. Here is how I would do it. Assume $a=b$. With Axiom 17 and ...
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### Presentations of the theory of categories

The classic way to encounter the theory of categories is via Set Theory via the typical definition we see for categories. We see all kinds of categories that are equivalent to the category of small ...
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### Can we strengthen the Maximize principle as to completely rely on it in determining the truth of set theoretic statements?

The question is about the truth of opposing set theoretic statements extending $\sf ZFC$, that are equi-consistent with $\sf ZFC$? If we adopt the "universe" view as the background for ...
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### About the presumed existence of experimental mathematics in deep learning

My question is rather simple. I am aware that the site "prefer questions that can be answered, not just discussed.", but I do not know where else to ask my question, and I should be glad if ...
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### Simultaneous / blended develop logic and set theory? [duplicate]

My goal right now is to gain a deep understanding of how to talk about mathematical objects formally. The presentation of how to do this in most books is generally to "assume some basic set ...
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### Concise list of ZFC Axioms for beginners

I am teaching a course and want to provide students with a simple explanation of the ZFC axioms without technical jargon. I try to define most of the primitive words in the list with the following ...
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### What is the Turing degree of the set of True formula of Arithmetic whose order is an infinite ordinal

This question was originally posted as a part of this other question, but I was suggested to make a new question for this part. In the first question I asked about the Turing degree of the set of ...
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### Mathematical object described irrespective of a foundational system?

Is there a mathematical discipline that studies mathematical objects based on their behavior rather than their encoding? I ask because a group is classically ...
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### What is the Turing degree of the set of true formula of Second Order Arithmetic?

The set of true formula of First Order Arithmetic is not arithmetical (by Tarski's undefinability theorem) and it has Turing degree $\emptyset^{(\omega)}$. What about the set of true formula of ...
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### How is self-reference usually handled in math?

Define statements A and B: A: “this statement” is Y B: “this statement” is X Assume I may combine statements using the logical operator “and”. Define statement C: C: A and B. (Meaning “C asserts ...
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### Proof of the existence of $\varnothing$ and $\{\varnothing\}$ [closed]

I would like to know a proof of the following statements from pure ZF axioms: $\exists x\forall y (\lnot y\in x)$ $\exists x\forall y [y\in x \Leftrightarrow\forall z (\lnot z\in y)]$ The reason ...
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### Purely geometric derivation of the circle surface

There are ways to get the surface or circumference of the circle by considering the areas and lengths of inscribed or Inscribing polygons. However this relies on using trig functions. This is ok at ...
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### Single-sorted (arrows-only) definition of infinity-category

It is possible to give a single-sorted (arrows-only) definition of a 1-category. For instance, see this nlab page. The basic idea is to identify objects with their identity morphisms. Is it possible ...
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### Is every math statement exponential in worst-case to prove?

 All math axioms can be expressed as grammar rules like the following: A -> B (directed) A <-> BC (undirected) See ...
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### On the meaning of the Axiom Of Regularity in terms of “ repeatedly taking the union” .

"whenever I try to chase down a chain of members, it must stop at some finite stage. You can think of it in this way. We have a string of sets $x_1,x_2,x_3...$ where each is a member of the preceding ...
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### Can we build set theory from category theory?

Can we get set theory from category theory? Or maybe can we consider both of them at the same time when building a foundation for mathematics? And also, I have read that almost every known ...
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### The legitimacy of topos theory and intuitionism.

This is an exercise in critical thinking. I am not looking, therefore, for opinions on the matter; rather: I would like to know the evidence (whatever that might mean). Background: I have a ...
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### Is Physics a Good Argument for Classical Math

(I posted this on philosophy stackexchange as well. Let me know if it belongs there more than here.) Is the success of classic mathematics in predicting the outcome of experiments in our physical ...
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### Set Theory Foundations book recommendation (meta-math)

I'm looking for a book on set theory foundations that goes into the metamathematics of it all. I worked through Kleene's Introduction to Metamathematics. In that text he proves godels incompleteness ...