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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).

2
votes
3answers
38 views

Proving Equivalence Relations, Constructing and Defining Operations on Equivalence Classes

I think I have an intuitive sense of how ordered pairs can function to specify equivalence classes when used in the construction of integers and rationals, for example. I put the cart before the horse,...
0
votes
1answer
38 views

Definition of < in Construction of Reals

In the section of Spivak's Calculus on the construction of the reals, < is defined: if alpha and beta are real numbers, then a < b means that a is contained in b (that is, every element of a is ...
3
votes
0answers
46 views

Model for the type-theoretic axiom of choice in Coq.

This is the request for references. It is a known fact that there is a model of ZFC in ZF, so ZFC is consistent if ZF is consistent. It is also know that there is a double-negation Godel-Gentzen ...
0
votes
1answer
53 views

Is it possible to prove Regularity with Transfinite Induction only?

Let us assume that we have only statement of transfinite induction. (And maybe some other well-know axioms) My question: "Is it possible to derive from it a regularity axiom as a theorem?". Some of ...
0
votes
1answer
81 views

Formal definition of a functor in $\mathsf{ZFC}$

Let $\mathcal{C}$ and $\mathcal{D}$ be categories. The first thing that comes to mind when considering a definition of a functor between $\mathcal{C}$ and $\mathcal{D}$ in $\mathsf{ZFC}$ is that it ...
1
vote
1answer
79 views

Category of functors $[C,D]$ and Grothendieck universes

A set $U$ is a universe if for any $x \in U$ we have $x \subseteq U$, for any $x,y \in U$ we have $\{x,y\} \in U$, for any $x \in U$ we have $\mathcal{P}(x) \in U$, for any family $(x_i)...
2
votes
1answer
54 views

Why we can safely treat objects like mathematical entities?

In the study of number systems we learn the axioms of real numbers. For example: The commutative axiom $X.Y = Y.X$ The distributive axiom $X.Z + Y.Z = (X+Y).Z$ Well, the thing is that we are ...
7
votes
6answers
718 views

How do we know our definitions don't lead to contradictions

How do we assure ourselves when defining an operation that it does not lead to contradictions? For example 0! := 1. I understand the practicality of why it is defined this way, but I am wary of what ...
-2
votes
1answer
61 views

Are there number systems that fix divide-by-zero? [duplicate]

Natural numbers are closed under addition and multiplication, but not subtraction. Fixed by... Integers are closed under subtraction, but not division. Fixed by... Rational numbers are closed under ...
2
votes
2answers
49 views

Translate the following English sentences into symbolic sentences with quantifiers.

this is my solution of my homework, is that true 100%? Please if there is any mistake tell me because my professor is so careful. Thanks. (Sorry, I don’t speak English well) i) Translate the ...
1
vote
0answers
73 views

Is everything a set? [closed]

ZFC set theory has the surprising property of being able to represent many mathematical objets that have intuitively nothing to do with sets. For example, a couple is represented by a Kuratowski pair, ...
3
votes
0answers
48 views

What part of arithmetic can be founded on recursive functions and without unbounded quantification?

Reading Skolem's 1923 Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlicher mit unendlichem Ausdehnungsbereich (Foundation of elementary ...
1
vote
1answer
67 views

Uniqueness of the empty set

When reading about set theory, I came across a simple question about the "uniqueness" of the empty set: Let $\mathfrak{M} \models ZF$, let $M$ be the domain of $\mathfrak{M}$. Let's create an inner ...
2
votes
1answer
24 views

Using this definition of ordinals, do I need foundation?

Definition. A set is an ordinal if it is a transitive set of transitive sets. This is the simplest definition of an ordinal I have ever encountered, and I happen to like it a lot for this reason. ...
0
votes
0answers
62 views

ZFC and Banach Tarski paradox [duplicate]

Why do we believe in ZFC so much and study it so hard, when it already yields something so unreasonable as is the well-known Banach-Tarski paradox?
-1
votes
1answer
25 views

Is applied statistical mathematics also can be explained by the zfc set theory?

I know that here there is already a lot of explanations about the zfc/zf/aca axioms but i wanted to ask if hypothetically people realy wanted to explain applied statistics to a creature that only ...
1
vote
2answers
39 views

How does the well-orderedness of the set of natural numbers follow assuming the inductive-set definition of natural numbers

Assume that $\mathbb R$ is an ordered field (i.e. $\mathbb R$ is a model of real numbers). We define the set of natural numbers $\mathbb N$ as the smallest inductive set containing $1_\mathbb R$ (...
6
votes
3answers
692 views

Why is the principle of explosion accepted in constructive mathematics?

I think something is wrong with the principle of explosion, because according to it, if I know $P\wedge \lnot P$, I can deduce $Q$ though I don't know anything about $Q$. Is it really constructive to ...
-1
votes
1answer
44 views

What makes a structured set a set of numbers?

I wonder which characteristics of a (structured) set $X$ let us consider it a set of numbers. Does the existence of two operations $+: X \times X \rightarrow X$ and $\times: X \times X \rightarrow X$...
-2
votes
1answer
32 views

manners of collection [duplicate]

I encountered idea of considering Sample space $\Omega$ as collection of function so far as I saw. Can you show the proof of thesis $x \neq \{x\}$? In a book I saw this thesis is included in context ...
3
votes
1answer
62 views

How to prove $\sqrt{2}\in \Bbb{R}$ with Dedekind cuts?

Problem statement: Prove that $\sqrt{2}\in\Bbb{R}$ by showing $x\cdot x=2$ where $x=A\vert B$ is the cut in $\Bbb{Q}$ with $A=\{r\in\Bbb{Q}\quad : \quad r\leq 0\quad \lor \quad r^2\lt 2\}$. Denote the ...
0
votes
0answers
42 views

Why is the axiom of choice necessary? [duplicate]

For long time, I cannot appreciate the axiom of choice: $\forall \lambda \in \Lambda (A_\lambda \ne \emptyset) \Rightarrow \Pi_{\lambda \in \Lambda} A_\lambda \ne \emptyset$. For any $\lambda \in \...
1
vote
1answer
96 views

Existence of Natural Numbers as an Axiom

By natural numbers $\mathbb{N}$ I understand any set satisfying Peano axioms: $0 \in \mathbb{N}$ $\sigma : \mathbb{N} \to \mathbb{N}$ $\forall n \in \mathbb{N} \; . \; \sigma(n) \neq 0$ $\forall n,m \...
0
votes
1answer
40 views

Transfinity of $\mathbb{Q}$ and $\mathbb{Z}$ [closed]

I have read in here: https://math.stackexchange.com/a/2899795/588260 that $\mathbb{Q}$ is 'larger' than $\mathbb{Z}$. I assume it has to do with transfinities in which the one of $\mathbb{Q}$ has a ...
0
votes
0answers
31 views

Subtyping of Prop in Coq. Implementation of Ackermann class theory. First-order theories.

I am trying to implement Ackermann set theory. The first approach is the code below. But there is an incorrect axiom "rs_ax". That's because for every x formula (F x) shouldn't contain predicate M. $\...
1
vote
1answer
41 views

$n$-tuples as nested ordered pairs: a formal definition using recursion on $\mathbb{N}$

Using conventional set theory as a foundation, there are two most popular definitions of an $n$-tuple of elements of $X$: A function $n\to X$. For any $x_1,...,x_n \in X$, $(x_1,...,x_n, x_{n+1}) = ((...
23
votes
6answers
3k views

Is consistency an axiom of mathematics?

I watched the numberphile video on Gödel's Incompleteness Theorem today, and I was wondering about something. It seems the key to accepting the truth of Gödel's Theorem is to demand that mathematics ...
17
votes
1answer
301 views

A model-theoretic question re: Nelson and exponentiation

EDIT: I am not asking about the validity of exponentiation, or PA. My question is about a specific technical claim which Nelson makes in this article (pp. 9-12): that a certain theory does not prove ...
3
votes
2answers
112 views

Foundations of Mathematics - Where to Start?

I am interested in the foundations of mathematics. However, I don't know where to start. From what I've read, the most popular foundational system for mathematics is Set Theory: ZFC specifically. ZFC ...
3
votes
1answer
97 views

Is Godel's incompleteness theorem unavoidable?

So after Godel's Incompleteness theorem and the fact that some theorems mathematicians are interested in are independent of ZFC (e.g. Continuum Hypothesis) is there some hope for some other ...
0
votes
2answers
103 views

Why Set Theory as Foundation? [closed]

Set Theory, particularly ZFC, is the most widely accepted foundation for mathematics. Why is that? Why is Set Theory (and, in particular, ZFC) the (in some sense) "best" foundation we can come up ...
6
votes
2answers
227 views

Categoricity of second order theories - precisely what does it mean?

As far as I am aware, it's been proven that the second order theories of $\mathbb{N}, \mathbb{Z},\mathbb{Q},\mathbb{R}, \mathbb{C}$ are categorical. I am sure that this is the case at least for $\...
3
votes
3answers
578 views

Is set theory used in proofs of theorems in logic?

Is set theory used in proofs of theorems in logic ? For example, is set theory used in Gödel's incompleteness theorem ? If yes, than does it mean that if we use other from ZFC set theory or even other ...
0
votes
0answers
16 views

2D plane characterization for Euclidian space of >1 dimension

Does the following characterization of a 2D plane hold for all Euclidian spaces of >1 dimension? Plane: a two-dimensional continuum of points with unbounded area defined by two intersecting lines ...
0
votes
1answer
32 views

Where can I find a worked example of Birkhoff's Third Postulate?

I've been reviewing unpackings of Birkhoff's postulates and I may be out of my depth, at least as far as terminology is concerned. I think that I understand the general concept of being able to ...
2
votes
3answers
195 views

Does countable induction over $\mathbb N$ require the axiom of infinity?

Consider ordinary induction over $\mathbb N$: Proving $P(0)$ Assume $P(k)$ being true for some natural $k$ Prove $P(k+1)$ Does this require the existence of $\mathbb N$ and hence the ...
2
votes
1answer
43 views

Mathematical objects

If the essence of mathematical objects isn't important to mathematicians but rather what they do and how are they related is there a branch of science or mathematics itself that examines exactly this?
2
votes
0answers
43 views

Non-set-theoretical foundations

Nowadays most ideas of foundations are based on some set theories. But are there some non-set-theoretical foundations, I mean are there some ideas of creating a theory which can foundate other math ...
0
votes
1answer
80 views

What is Mathematics? [closed]

I study electronic engineering at university, 3rd course. I had to use mathematics a lot, from basic algebra to analysis. Yesterday, after watching some mathematics-related videos and reading some ...
2
votes
1answer
204 views

Why does, in theory, you can prove Riemann hypothesis with Godel Incompleteness Theorem, but can't prove the consistency of mathematics?

Some time ago, I watched this fascinating episode of Numberphile about Godel Incompleteness Theorem: https://www.youtube.com/watch?v=O4ndIDcDSGc&t=4s In the video, Professor du Sautoy explains the ...
2
votes
0answers
82 views

Interdefinability of set, type and category theories [closed]

There seem to be, broadly speaking, three1 distinct foundations of mathematics: set, type and category theory (the latter as per Lawvere), in which it should be possible to formalize all mathematics ...
6
votes
4answers
120 views

Does the successor function imply order?

My confusion is how order arises from the Peano axioms (wikipedia link). From this question I'm not sure that "successor" means "greater than." It seems you could take $\mathbf{0}$ and then the ...
1
vote
1answer
49 views

How to prove axiom of pairs from power set axiom and replacement?

This the exercise 2 of section 6 of chapter 1 of "Set Theory: An Introduction to Large Cardinals", by Frank R. Drake: Show that A4 can be deduced from A3, and that A3 can be deduced from A9. Show ...
2
votes
0answers
47 views

Do we need to go in meta-theory for proving completeness?

Do I need to go in some meta-theory A for proving theory B is complete ? Or can I do it inside the theory B ? If I really need it, what is the reason for that ?
2
votes
1answer
85 views

Metaphysical/psychological aspects of describing a formal language (mentioned in Bourbaki)

In the introduction to Bourbaki vol. 1, we read: "It goes without saying that the description of the formalized language is made in ordinary language, just as the rules of chess are. We do not ...
3
votes
1answer
147 views

Why is Axiom of Pairing needed?

I'm learning ZFC set theory and I'm very confused about the Axiom of Pairing. Axiom of Pairing: For any $a$ and $b$ there exists a set $\{a, b\}$ that contains exactly $a$ and $b$. It seems that ...
2
votes
1answer
164 views

Are there any new axioms being developed?

Are any mathematicians working on finding new axioms, either for ZFC or another foundational theory of math? I know that because of Godel's incompleteness theorem, it's impossible to construct a ...
2
votes
1answer
54 views

Choosing axiom schemes for a logical theory

In a Hilbert system, there are many ways that we can choose axiom schemes. My question is: 1- How do we know that we have defined enough schemes? What would happen if I remove a scheme from the list? ...
0
votes
1answer
41 views

To what part of propositional grammar does “+” belong?

This is a follow-up to Is "+" a two-place predicate? If the function symbol "+" is not a "two-place predicate", then what part of mathematical speech is it? The context is the "Logical ...
1
vote
2answers
62 views

Is “+” a two-place predicate?

In Volume I of Fundamentals of Mathematics, Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle, it is stated that "[T]he function sign '+' is a two-place predicate". To me, a ...