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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).

Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague.

Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics.

The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.

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exercise in Tao analysis I book

I am supposed to prove that if a is a positive natural number then there exists exactly one number b, such that the increment of b is equal to a. My idea was to induct from the base case a = 1, but I am I can't proceed this way because I haven't yet…
Adam
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What's the difference between the Well-ordering principle and Least Upper Bound property?

Well-Ordering Principle : Every non-empty set of positive integers contains a least element Least Upper Bound Property: Every nonempty bounded subset of the real numbers has a least upper bound. Is the Well-Ordering derived from the LUB property?
Lemon
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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 infinite sets. Let $X_1 = \{{x_1}_1, {x_1}_2,…
subscript42
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Euclid's Divison Lemma and Mathematical Logic

While I was trying to prove Euclid's Divison algorithm, I came across a certain problem. Let me share the proof and the problem. Lemma : Let $ a , b \in \mathbb{N} , a > b $ then there exists $q , r$ , $ a = bq + r , 0 \leq r < b$ Proof : Let $S$…
user683949
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Is my proof about the commutativity of multiplication correct (Tao lemma 2.3.2)?

I have already read the answer to the following question. I have a different proof and I don't know if it is correct. The textbook also doesn't provide solutions and I am trying to self-study (I am asking questions here so I guess its not really…
TheGeometer
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To define a set theory in the framework of axiomatic euclidean geometry? I do not of course mean the whole ZFC...

Is it possible to define a set theory, of course weaker than ZFC, in terms of axiomatic euclidean geometry eg Hilbert (corrected Euclid). My idea is that once we shall have defined such a set theory we could be able based on this to define a part of…
user3687245
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What is foundations of mathematics ? And why isn't it taught more often?

I have just come across "Foundations of Mathematics". Wikipedia describes it as Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form…
ng.newbie
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Can ZFC be taken as the foundation of mathematics without first-order logic?

On the wikipedia page for ZFC: link It says that ZFC is formally a theory in logic. And I guess all of mathematics relies on logic, but we use logic informally when creating mathematical arguments.(This seems to be done in all textbooks.) Does this…
user119615
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A foundation where objects are unlabelled

Okay, this may seem a crazy question. If it does, it's because I've been thinking too much about foundations, recently. I've read a bit on ZFC, on the need for universes for categories, on HoTT (though not far in that region), and on ETCS. I…
Nethesis
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A question on Mortiz Pasch works in foundation of mathematics

I am reading "Introduction to the Foundation of Mathematics" ,by R L.Wilder(2nd Ed.), where Mortiz Pasch's works are described in paragraph 1.5. There is a quotation in that paragraph- " For if,on replacing the geometric terms in the basic set of…
Shubhadip Ghosh
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Can a metatheory recognize objects from all permitted axiomatic systems?

Can there be/ has there been created a meta-theory that recognizes the existence of all objects whose existence is derivable from (or just "true in") each axiomatic system analyzable by the metatheory? From my (naïve) position, it seems like you…
Jacob Wakem
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How do you prove that $m\in\Bbb Z$ implies that $m^2$ is even with ZFC?

On this page is a very simple theorem (Theorem 1.1) $$\text{If }\;m\in\mathbb{Z}\;\text{ is even, then }\;m^2\text{ is even}$$ How would you prove this theorem using ZFC set theory? What would it look like? How is the theorem represented using…
sphere100
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Bijection between Power Set of Natural numbers and Power Set of integers

There is for sure more than one method to do this , however I want to understand the hint provided in my book .The book says to use the bijection between N and Z given by f(n)=n/2 if n is even and f(n)= (1-n)/2 if n is odd .I think I can prove one…
herashefat
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Help understanding Dedekind cuts. I would like to build a good foundation to be able construct proofs

So far what I know about Dedekind cuts are 1. it contains a rational number but does not contain all rational numbers 2. every rational number in the set is smaller than every rational number not belonging to the set. 3. it does not contain a…
Joesixpack
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Given that the set of all proofs is not bounded, how come we can do math at all?

If we search for the proof of a theorem then, assuming that there exists a proof, it lies in the set of all proofs of all provable theorems. But since this set is infinite, this means that the shortest length of the proof can be arbitrarily large.…
Count Iblis
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