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).
1,361 questions
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Would there be purpose of goal of setting equations? [closed]
If there is a equation denoted like so,
$${\displaystyle x^{5}-3x+1=0}$$
Negelecting the meaning of this equation or solving it, would there be historical background equations were adopted or could ...
- 117
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1
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Set-theoretical conventions in Pedersen's Analysis Now
In chapter 1 of Gert Pedersen's Analysis Now (specifically the exercises), when dealing with "collections" of proper (equivalence) classes, one avoids standard set-theoretical difficulties ...
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Is there a definition for stronger independence of a proposition concerning an axiomatic system.
I wonder if a definition like the following already exists.
If we have an axiom set $A$, can we define the independence of $\alpha$ in the situation where we are adding a new axiom $\alpha$ to $A$ ...
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Reference requests for a survey of mathematical theories of truth in depth?
I am very interested in theories of truth at the moment.
For example, I am aware of several different kinds of semantics for logic. There are the semantics provided by model theory, the semantics ...
- 1,262
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Does truth not exist in ZFC, or is it merely not definable?
I am having a lot of trouble with the concept of Tarski's undefinability theorem as it relates to set theory.
Tarski's undefinability theorem says that there is no formula $Tr$ on the natural numbers ...
- 1,262
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1
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Proving arbitrary products exist in a Mac Lane universe
I am going through some leftover exercises from Mac Lane's Category Theory for the Working Mathematician. I am currently dealing with the foundational chapter, in which the concepts of small and large ...
- 2,584
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answer
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What real numbers correspond to second order logic definable Dekekind cuts?
This question spurred from a thought I had: does every (lower) Dedekind cut have a (finite) second order logic formula that defines it?
Fix the usual setting: the domain is $\mathbb{Q}$ with the order ...
- 2,584
12
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2
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What type theory really is?
I just wanted to figure out what type theory is, especially dependent type theory (I'm interested in how they corresponds to locally cartesian closed categories), but I just didn't find any definition ...
- 706
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To "enrich" as opposed to "dissolve" the point?
Grothendieck dissolved the classical notion of a point into a functor of points
$$
h_X : (\mathbf{Sch})^{\mathrm{op}} \longrightarrow \mathbf{Set},
\qquad
h_X(S) = \mathrm{Hom}(S, X),
$$
re-...
- 1,209
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Size classification for Categories
I am interested in the foundational aspects of Category Theory, in particular size issues. I was looking for some sort of taxonomy but it seems even the nLab does not have too many entries on this ...
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With ZFC foundations, if all math objects are sets, where do these sets "live"?
Let's say we're doing ordinary mathematics, and we want ZFC to be our foundations, such that all of our mathematical objects are sets.
I have long had the idea in my head that these sets that ...
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2
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If infinity isn’t a number, why can we still do arithmetic with it in calculus limits?
I understand that infinity is not a real number, but calculus uses expressions like “x → ∞” or “1/∞ = 0.” What’s the rigorous difference between using ∞ symbolically in limits versus treating it as a ...
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What is a model of a first-order language exactly? [duplicate]
A model of a first-order language is an ordered pair that contains a universe and a related interpretation function for predicate letters, function and constant symbols. That function and universe are ...
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Set Theoretically, NOT CONCEPTUALLY, are Position Vectors the same as Points
TO BE CLEAR: I am asking from a mathematical purist, set-theoretic, construction of math point-of-view, not an applied point of view.
Does $(1,2,3)=\langle1,2,3\rangle$?
If we disect a euclidean ...
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1
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Harvey's Friedman work on Con(SRP) [closed]
Here on the page 9 what does he mean by
this notation:
Recall our fixed $k \geq 1000[3]$ and below by $L[k[1],k[1],k]$.
I cannot find a definition in the paper.
