Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

0
votes
1answer
32 views

Frege's argument for the existence of abstract mathematical objects

I have some trouble understanding Freges argument in particular as presented here, https://stanford.library.sydney.edu.au/entries/platonism-mathematics/#FreArgForExi In particular the first premise i....
1
vote
0answers
29 views

Has there been any mathematical research relating to the natural semantic metalanguage theory?

This topic was called to my attention in an answer to my question in the philosophy community. https://philosophy.stackexchange.com/q/63580/35343 Though mathematics is not a "natural language", the ...
0
votes
1answer
70 views

About a famous assertion by B. Russell on mathematical truths considered as conditional truths. Is this claim also true of axioms?

In Mysticism and Logic, Russell says that : "Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another ...
3
votes
2answers
127 views

Plurality of arithmetics? Or absoluteness of arithmetical truths? (i. e. Are all mathematical truths actually conditional?)

The following assertion is attributed to Russell ( as a quote from Mysticism and logic) : Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is ...
0
votes
1answer
57 views

Developing new theories by transfinitely iterating the Godel sentence construction

In Turing's Ph. D thesis "Systems of Logic Based on Ordinals", he writes of a simple way to use Gödel's incompleteness theorem to devise a transfinite sequence of new theories. The sequence proceeds ...
0
votes
1answer
48 views

Number system construction book recommendation

I am coming to the end of 'a logical introduction to proof' by Cunningham and was thinking of continuing with some foundational topics. As such i think i may try the following: Set Theory: A First ...
2
votes
0answers
53 views

Definition of the set $\mathbb{Z}_+$

I'm currently going through the introductory chapters of Munkres' Topology, and in chapter 1 section 4 of the second edition, Munkres attempts to briefly establish some Mathematical foundations for ...
1
vote
1answer
67 views

What is in $\mathbb{I} - \mathbb{N}$

In the Wikipedia page for the axiom of infinity it states that the natural numbers are a subset of the set that the axiom defines. Then gives a way of extracting the numbers from it. However I'm ...
1
vote
0answers
42 views

A consistent ZFC implies that ZFC has a model that is not well-founded [duplicate]

I've come across a problem in a book about the Zermelo–Fraenkel set theory I'm reading and am kind of stuck: Assume ZFC to be consistent. Show that ZFC has a model that is not well-founded/regular. ...
0
votes
1answer
65 views

Do Gödel's incompleteness theorems apply to finitist axiomatic systems that reject the axiom of infinity?

I'm curious which axiomatic frameworks Gödel's incompleteness theorems apply. Do the theorems apply to any axiomatic system that adopts potential infinity? Or just to those that adopt completed ...
0
votes
1answer
24 views

How would I show that PA proves the godel sentence for PA implies Con(PA)

Started off with definitions: godel sentence for PA is the sentence in PA that cannot be proved nor disproved Con(PA) formalizes PA is consistent We know godels first incompleteness theorem is ...
0
votes
0answers
50 views

Why is the calculus of constructions called that way, and what is a “construction” in CoC?

I'm reading about the calculus of construction Nederpelt & Geuvers' book "Type theory and formal proof". I can see that CoC allows us to extend the curry howard isomorphism from simply typed ...
3
votes
1answer
96 views

Can we prove the Peano axioms from a type theoretic construction of the natural numbers?

Here are two quotes that, while not literally contradictory, reach conclusions that are opposite in spirit. The first one states that the Peano axioms can be proven to hold for an explicit ...
8
votes
1answer
60 views

Comment about type theory in Lawvere and Rosebrugh's Sets for Mathematics

The Foreword to Sets for Mathematics (second section, titled Organization) contains the following comment, about the differences between ETCS and other foundations of mathematics: Each map needs ...
1
vote
0answers
46 views

Replacing differentiation with anti-integration?

When thinking about the proof of the differentiability of Taylor series, I noticed that the theorem was proved by using properties of integrals. This got me thinking: To what extent can the role of ...
-1
votes
1answer
37 views

Predicates and mathematical objects

I'm reflecting about mathematical objects as numbers, sets/classes, graphs and so on. Any class correspond to a predicate in one variable and any graph correspond to a predicate in two variables. ...
0
votes
1answer
62 views

If $m*n$ divides k, then both m divides k and n divides k

How might I complete this proof? Let $(m*n)p = k$ such that p, m, n $\in$ Z. Then $m(n*p) = k$ and $n(m*p) = k$ thus n divides k and m divides k since $n*p$ and $m*p$ must also be integers. ...
0
votes
2answers
25 views

Proof by Induction: Recursively Defined Sequential Set

We recursively define a sequence of subsets of $\mathbb Z$ as follows: Let $S_0=\{0\}$, and let $S_{n+1}=\{2m: m \in S_n\} \cup \{2m+1: m \in S_n\}$ for all $n \geq 0$. (So $S_1=\{0,1\}$, $...
0
votes
2answers
54 views

Prove that $\forall \epsilon>0, \epsilon>a \implies 0 \geq a$

I am doing a course on basic real analysis in which firstly i am emphasising on real numbers. My book says that real number satisfies the following axioms. 1)Field Axiom 2)Extend Axiom 3)Order Axiom ...
1
vote
0answers
55 views

Nonstandard proof codes for Con(PA)

Note: This question is inspired by a paper being discussed in the FoM mailing list. The formal statement Con(PA) is a statement about the existence of a proof code for the derivation of a ...
1
vote
1answer
90 views

Why do mathematicians generally write definitions in “declarative” rather than “imperative style?

In programming, we can make the distinction between declarative / functional and procedural / imperative programming. The distinction is not exact, but nevertheless meaningful. One major difference ...
8
votes
0answers
64 views

Natural examples of “set-theorems” provable in MK but not ZFC and NBG

NBG set theory proves nothing additional about sets relative to ZFC. MK set theory, on the other hand, is stronger, meaning it should prove statements about sets that NBG and ZFC don't. Are there ...
0
votes
0answers
42 views

Finite axiomatization of second-order NBG

First-order NBG set theory is finitely axiomatizable. The proof of this basically shows that the axiom schemas, in the presence of the other axioms, can be reduced to a finite set of cases, and are ...
1
vote
1answer
88 views

Definitions in metamathematics

In model theory, the satisfiability relation $ \vDash$ between a model $M= (D,f)$ and a set of formulas tells us when a formula $\varphi$ is true or not in the model ("interpretation") $M$. This ...
0
votes
1answer
68 views

Why is the Axiom of Well-ordered Choice not strong enough to prove Zorn's Lemma?

This is based on this question: How strong is the axiom of well-ordered choice? The "axiom of well-ordered choice" says that any transfinitely-indexed family of sets has a choice function. The ...
11
votes
1answer
186 views

How strong is the axiom of well-ordered choice?

I sometimes see references to the "Axiom of Well-Ordered Choice," but I'm not sure how strong it is. It states that every well-ordered family of sets has a choice function. By "well-ordered family," ...
1
vote
1answer
97 views

What is the most expressive logic?

What is the most expressive logic studied in the literature (in terms of expressing properties about a structure in the sense of model theory)? Many people talk about second-order logic, but third-...
1
vote
1answer
87 views

Is there a notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$?

It is accepted belief $\mbox{countable }\infty$ is not $\mbox{uncountable }\infty$. Is there notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$ (latter ...
2
votes
1answer
96 views

Formalizing the deduction theorem in the metatheory

Here is the deduction theorem, in the "$\Leftrightarrow$" version (I'm considering it for first order logic): $$\Delta \cup \lbrace A \rbrace \vdash \lbrace B \rbrace \Longleftrightarrow \Delta \vdash ...
-2
votes
1answer
72 views

Would there be any consequences in accepting conjectures that are essentially true? [closed]

There are many unproven conjectures that if you heuristically took a probability of it being true, it would basically be almost $100\%$ true. I can see why mathematics must be rigorous as many ...
4
votes
0answers
284 views

Metamathematics and the foundations of mathematics

I have some really big doubts about what is the real starting point of all (formal) mathematics. For example: when I search on internet or study texts about the foundations of mathematics such as ...
1
vote
2answers
320 views

What are the other foundations of math?

I know there are at least three foundations of math: Zermelo-Frankel set theory. (Sp?) ECTS Category theory Are there any others? Experimental foundations are welcome.
3
votes
2answers
185 views

The Scope of Axiomatic Set Theory

I am currently studying ZF set theory in terms of first-order logic and I am having trouble understanding the motivation behind this axiomatic formulation of set theory. ZF set theory is a first-...
0
votes
0answers
145 views

How did mathematicans prove things before ZFC or Peano

I do not get it. How was a+b=b+a or a (b +c) = ab + ac proved by ancient mathematicians before set theory or Peano axioms? If you cannot logically prove the laws of arithmetic without Peano or set ...
0
votes
0answers
40 views

Natural Transformation Between Covariant Hom-Functors

Let $\mathscr C$ be a category and $A,B\in Ob_\mathscr C$. I don't think it matters, but assume that $\mathscr C$ is locally small. I want to find a natural transformation between the covariant Hom-...
5
votes
1answer
220 views

Formalist understanding of metalogical proofs

How can a formalist understand a metalogical proof such as the completeness theorem? These proofs exist exclusively outside of a logical system where the "rules of the game" are undefined. These sorts ...
0
votes
0answers
82 views

Can you recommend literature - easy/gentle/for self-study/introductory… - for the following topics…?

I am looking for literature that is as self-explanatory, easy, gentle, readable to the beginner, suitable for self-study, etc.. as possible, in the following fields. (I mean the mathematical part as ...
1
vote
1answer
149 views

Legitimacy of Consistency Proofs

In this question I asked yesterday I put forward two interpretations of a statements such as "System X is consistent". (a) we can think of it as saying no finite sequence of applications of logical ...
3
votes
1answer
112 views

Consistency of PA from a Formalist Perspective

In this lengthy thread there's people bickering back and forth about the consistency of PA (Peano Arithmetic) and misunderstandings abound. In reading it I came to an understanding I found useful, ...
1
vote
2answers
170 views

Well-definedness of uncomputable functions.

I have been reading about Rayo's function and uncomputable functions in general, and have gotten very confused. There is apparently concern over the well-definedness of Rayo's function, but I never ...
1
vote
2answers
37 views

What function $f(n)$ is defined by $f(1)=2$ and $f(n+1)=2f(n)$ for $n\geq 1$

I have to 2 qusetions in a mathematical induction homework: 1-What function $f(n)$ is defined by $f(1)=2$ and $f(n+1)=2f(n)$ for $n\geq 1$ My attempt: $f(1)=2$ $f(2)=2f(1)=2(2)=2^2$ $f(3)=2f(2)=...
1
vote
1answer
78 views

Development of topology and differential geometry in ETCS

I would like to ask for some reference textbooks or articles, or any information that you know about developing topological concepts and differential geometric concepts using as a foundation ETCS ...
6
votes
1answer
100 views

What is the Turing degree of Truth?

First of all by Truth I mean the set $T$ of the Gôdel numbers of the true formulas of first order arithmetic. First order arithmetic is not decidable and $T$ is not decidable, additionally the ...
1
vote
1answer
98 views

Hilbert Program: Consistency vs Soundness

I have been wondering why exactly Hilbert asked for a decidable, complete, and consistent set of axioms for mathematics, rather than a decidable, complete, and sound set of axioms. Consistency being: ...
0
votes
2answers
60 views

How to define substitution using ZFC

One question I've had regarding ZFC is how to define substitution. I cannot see how it's possible, despite the frequent use of substitution within both pure and applied mathematics. Just to be clear, ...
1
vote
1answer
41 views

Question on the reasoning behind determining surjectivity of a function

I understand the idea of surjectivity and its definition: "A function f is surjective if f:A->B if $\forall y\in Y,\exists x \in X$ such that $y=f(x)$" However I have a question on the following ...
1
vote
0answers
63 views

Limit of a sequence related to non-well founded set theory [closed]

Consider the following: $$\alpha_0 = \{\} $$ $$\alpha_1 =\{\{\}\}$$ $$\alpha_2 =\{\{\{\}\}\}$$ $$...$$ Clearly $\alpha_i$ is considered a set for all $i \in \omega$. Then consider $ \alpha=\lim_{i \...
0
votes
2answers
67 views

Need help with proof by contradiction of implication.

Question reads: $$\forall n \in \mathbb{N}, \forall m \in \mathbb{N}, (n^4 + n^2 + 1 \ne m^2)$$ In english, this reads: for every n and m in the set of natural numbers, $n^4 + n^2 + 1$ does not ...
1
vote
0answers
25 views

Sequence of Number System Construction

After constructing the naturals, why construct integers before rationals? Is there a historical explanation? Couldn't ordered pairs of fractions constructed from the naturals be used to represent ...
1
vote
1answer
73 views

What is the difference between the logical formula $\alpha$ and expression “$\alpha$ is true”?

What is the difference between the logical formula $\alpha$ and expression "$\alpha$ is true"? I feel that there is a difference between those expressions. I think this difference is the difference ...