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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61 views

Would this argument support logicism?

Notice the following theory written in multi-sorted first order logic with equality and membership, with axioms of: Comprehension:$\small k=1,2,..,\omega, \omega+1,..$ $\forall x^{i_1},...,x^{i_n} \...
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120 views

What is the difference between 1 + 1 and 0 + 2

Elementary arithmetic tells us these two expressions are equal. Here what puzzles me about this. $1 + 1$ is symmetric in the sense we have two identical copies of the same entity (unity) and $0 + 2$ ...
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Finite sets and principle of excluded middle

This question is about the principle of excluded middle and its relation to the axiom of choice on finite sets. I am new to the principle of excluded middle and I was reading it on nLab and Wikipedia. ...
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Is the following provable in Zermelo–Fraenkel Set Theory?

Zermelo–Fraenkel Set Theory is a system of axioms for describing set theory. For example, Zermelo–Fraenkel Set Theory says things like: For any set $x$ and any set $y$ there exists a set $z$ such ...
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What is the mathematical symbol for “nothing”?

The question might sound weird. But I have a situation coming up while writing a research paper. I will try to put it simply. I want to define a random variable $X$ which takes the values from the set ...
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What do Logicians mean by a set? [duplicate]

I'm trying to learn some logic to understand different kinds of foundations of mathematics. However, most of the logic texts I've seen will define things like formal languages, valuations and models ...
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Meta-logic of Hilbert-style propositional calculus

I am trying to study the foundations of mathematics from the bottom-up (propositional logic then predicate logic then the axioms of set theory.) Currently, I'm considering the following Hilbert-style ...
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Set Theory internal to other foundational systems

There has been a push recently (at least in some circles) to change the foundations of mathematics from Set Theory to Type Theory (or some kind of Category Theory, etc.) I am interested in set theory,...
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Would mathematics based on lists obviate the need for the axiom of choice?

I'm trying to wrap my head around the axiom of choice and its equivalent well-ordering theorem. Imagine a mathematics founded on ordered lists rather than sets. So by construction, wouldn't every ...
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Equivalence or Isomorphism of Types

The context I'm working in here is dependent type theory used in proof formalization (in particular in Lean, though is likely not relevant). The question I have is best explained through examples. A ...
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111 views

Why not Hierarchy theory rather than ZFC be the foundation of sets and mathematics?

Hierarchy theory is the theory obtained by adding a one place function $V$ to the first order language of set theory. Define $``ordinal"$ along Von Neumann's, also $``<"$ is defined in the usual ...
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91 views

Definition of structure and satisfaction on Set theory(ZFC).

I'm studying mathematical logic with Enderton's book[1972]. Afther reading the definition of structures, satisfaction and models, I have thought that the definition really take set theorical notions ...
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What exactly are Classes?

As of now, I understand classes to be introduced for us to meaningfully talk about collections of object for which describing them with sets would create problems. I have a few questions about classes ...
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Is this a proper recursive ordinal notation for ordinals < $\omega^2$?

After making another question about ordinal notation I want to clear some confusion I have about the topic. Let consider ordinals less than $\omega^2$ (or in $\omega^2$) , any of such ordinals can be ...
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Good sources on the construction of mathematical definitions?

Please forgive sloppiness/ignorance, I have no formal background and am basically a fan. I've recently run into some problems by carelessly including statements of uniqueness and existence in ...
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A question on primitive notions.

This is a follow up question of sorts to my previous question at: On the notion of set equality. In ZFC, we have to consider a collection of primitive notions. In particular, we consider the notion ...
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140 views

Why is ZFC the foundational theory of mathematics?

In a prior posting here, the following is a quote from an answer: $\mathsf{ZFC}$ is so ridiculously overpowered that its inconsistency wouldn't really spill over to the rest of mathematics too ...
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On the notion of set equality.

I should start by providing some context. I want to understand the foundations of mathematics and in doing so have been looking at different texts involving ZF set theory and category theory. I ...
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Is there a set in ZFC that can not be obtained from Ordinals

Is there a set in ZFC that can not be obtained from ordinals (defined according the Von Neumann definition) via set operations (union, intersection, set difference) and power set?
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Foundation of mathematics for engineers [closed]

I'm a computer engineering student and I feel like I missed a lot in my studies, so I started learning everything from the beginning by myself, and I decided to start with the foundation of ...
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Are the axioms of ZFC set theory considered to be synthetic?

Pretty straight forward questions. I assume that ZFC would be considered a logicist program if its founding axioms were logical and analytic in nature, yet ZFC is often referred to as "extra-logical"....
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Prove Trichotomy and Multiplicative Ordering for Integers

Prove the following properties of $\mathbb{Z}$: (a) (Trichotomy) For any $x, y ∈ \mathbb{Z}$, precisely one of $x < y$, $x = y$, $x > y$ is true. (b) (Multiplicative ordering) For $x, y,z ∈ \...
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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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112 views

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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74 views

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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134 views

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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49 views

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?

[1] 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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How does the Axiom of Regularity apply to $A=\{1,2,3\}$?

I cannot understand the axiom of regularity. It says that any generic non-empty set $A$ contains an element $X$ such that $ X\cap A = \emptyset $. How can this be true? If, for example I have a set $A ...
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Can all math be done with axioms of length 3 or less?

[1] Assume math can be done with unrestricted grammars (See Are axioms in math equivalent to production rules in unrestricted grammars?). Can the grammar rules be rewritten to contain at most 3 ...
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Are axioms in math equivalent to production rules in unrestricted grammars?

In other words, the Curry–Howard correspondence is the observation that two families of seemingly unrelated formalisms—namely, the proof systems on one hand, and the models of computation on the other—...
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Example of how set theory is foundation for the rest of mathematics

I have heard it said that set theory can be seen as the foundation for the rest of mathematics. The article on set theory at brilliant.org put it as: Set theory is important mainly because it ...
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Characterization of V-stages in the absence of foundation or replacement

In the absence of foundation and replacement, we can say that $x$ is an ordinal iff $x$ is hereditarily well-founded (with respect to the relation $\in$) and hereditarily transitive, where $$ \text{$...
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139 views

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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162 views

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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69 views

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 ...
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152 views

Are we still Researching the Foundations of Math?

To my understanding, most of math can be founded in ZFC set theory. And, because of Gödel, we can never really prove the consistency of ZFC and hence mathematics as a whole. My question is, are we ...
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107 views

Proof of *2.38 in Principia Mathematica (Russell/Whitehead) [closed]

In Bertrand Russell and Alfred Whitehead's Principia Mathematica, they note that "The proofs of *2.37.38 are exactly analogous to that of *2.36." I have found a proof of *2.37, but would like to ...

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