Questions tagged [foundations]
Mathematical logic, Set theory, Peano arithmetic, Model theory, Proof theory, Recursion theory, Computability theory, Univalent foundations, Reverse mathematics, Frege foundation of arithmetic, Goedel's incompleteness and Mathematics, Structural set theory, Category theory, Type theory.
330 questions
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getting rid of existential quantifiers
It seems to me that for most of the twentieth century, axiomatic foundations for mathematical theories were constructed with the (mostly allied) goals of minimizing the number of primitive notions and ...
- 19.7k
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5
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Easy and Hard problems in Mathematics [closed]
Modified question:
I would like to know some examples of problems in Mathematics, for pedagogical purposes, which do not involve difficult techiques to solve the problem but with a change of context ...
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0
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418
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a priori grounds of mathematics [closed]
Hi,
from the title of the question you may guess I'm new to the site, and I am. Even though I've read the FAQ and I know that this isn't the place for such open questions (I believe it is open, but ...
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1
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Up-to-date version of Principia Mathematica?
Background: I found this interesting translation of Godel's On formally undecidable propositions of Principia Mathematica and related systems I that, along with translating it into English, uses more ...
- 1,052
10
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1
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Set-theoretical multiverse and foundations
I just had a look to the article The set theoretical multiverse by (mo user) J.D.Hamkins. Not being a logician and not knowing forcing techniques, I couldn't fully appreciate the mathematical ideas, ...
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7
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Is there a finite-dimensional vector space whose dimension cannot be found? [closed]
Is there a finite-dimensional vector space whose dimension cannot be found? Assume, we have somehow constructed a vector space whose dimension is finite, but yet unknown. Is there always an algorithm ...
- 441
8
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1
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Foundations: Existence of uncountable ordinals.
This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this ...
- 2,754
10
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4
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On a weak choice principle
[PLEASE SEE EDITS AT BOTTOM OF QUESTION]
Consider the following set-theoretic axiom:
For each set $X$ there exists a set-indexed collection $\{C_i \to X\}_{i\in I_X}$ of surjections such that for ...
- 35.4k
6
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1
answer
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How are mathematical objects defined from an ultrafinitist perspective?
I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...
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2
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832
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Intension vs. Extension: Coextensive relations in model and set theory
(originally posted at MSE as Same same but different: Coextensive relations in model and set theory, slightly modified)
The official definition of a structure in model theory in its presumably most ...
- 9,720
4
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2
answers
1k
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products in a category without reference to objects or sources and targets
Hi,
I was thinking about presenting categories with nothing but equations over morphisms. I wondered about products. The definition of a product has its genesis in the following diagram shape
A->B
...
- 1,313
4
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1
answer
605
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linear logic, diagrammatic calculus and foundations
Hi,
I have been interested in foundations for a while, especially categories as foundations. I am of the opinion that, as long as we present the theory of categories in SET, we will not be able to ...
- 1,313
2
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4
answers
1k
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Are inference laws consistent?
Please forgive me if this question sounds too naive... Well, in mathematics a formal theory consists of a collection of axioms $T$ (such as Peano arithmetics, or Group Theory, or ZFC), which ...
- 23.3k
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1
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Is it possible to construct a finite mathematical universe? [duplicate]
Possible Duplicate:
Is there any formal foundation to ultrafinitism?
Very recently I have come across the skeptic opinions of a school of mathematicians(ultrafinitist) over a physically ...
- 19
4
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2
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2k
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Dedekind's theorem
In "Was sind und was sollen die Zahlen?" Dedekind gives a noncircular
proof of the statement that a set is finite if and only if it cannot be
put in bijective correspondence with a proper subset. By "...
- 18.6k
1
vote
1
answer
332
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continuous maps between categories that are not functors
Hey,
Is it possible to define a map between two categories which preserves all products and binary equalizers and yet is not a functor, ie it does not satisfy one or more axioms of a functor? ...
- 1,313
7
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2
answers
736
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Sets as Combinatorial Games
Just a few days ago my seemingly eternal and recurrent fascination for Conway's combinatorial game theory (CGT) & surreal numbers had a recrudescence, so I grabbed this excellent survey, and began ...
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11
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Is PA consistent? do we know it?
1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs (...
7
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3
answers
3k
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incompleteness in real analysis
Godel's theorem tells us that any sufficiently powerful consistent formal theory of the integers is incomplete; but what about formal theories of the real numbers? More precisely, what about theories ...
- 19.7k
14
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1
answer
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Martin's "Philosophical Issues about the Hierarchy of Sets"
Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the ...
- 1,465
13
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7
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(Non?)-linearity of the consistency strength ordering in ZF
Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all ...
- 360
59
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8
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How true are theorems proved by Coq?
Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to ...
- 35.4k
74
votes
11
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12k
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Why hasn't mereology succeeded as an alternative to set theory?
I have recently run into this Wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set ...
- 5,902
11
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3
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Kunen's use of Countable Transitive Models
Hi,
I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, ...
8
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2
answers
863
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Consistent hierarchy of axiomatic systems
First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight.
I just learned in Sergey Melikhov's answer to another question ...
- 25.5k
5
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5
answers
1k
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Concrete models of abstract structures
Most mathematicians seem to be contented with the fact, that abstract structures cannot be directly modelled as such in a set theory without ur-elements. What seems to me the standard stance: Set ...
- 9,720
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4
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Subsystems of Peano arithmetic and incompleteness theorem
I think everyone is familiar with Goedel's incompleteness theorems. In particular they imply that PA (Peano arithmetic) can not prove its own consistency. Now my question is what is the largest ...
- 741
26
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1
answer
2k
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Nontrivial circular arguments?
There is a famous circular argument for the Prime Number Theorem (PNT). It turns
out that there exists an infinite sequence of elementary-to-prove Chebyshev-type estimates
that taken together imply ...
- 17.6k
2
votes
0
answers
167
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Multitype approaches to choice?
I wonder if anyone has developed a set theory which approaches the issue of the non-emptiness of products of non-empty sets via a hierarchy of types (comparable to how Von Neumann–Bernays–Gödel set ...
- 17.6k
17
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10
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Set theory and alternative foundations
Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set ...
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