All Questions
Tagged with set-theory mathematical-philosophy
117 questions
15
votes
5
answers
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Intended interpretations of set theories
In his Set Theory. An Introduction to Indepencence Proofs, Kunen develops $ZFC$ from a platonistic point of view because he believes that this is pedagogically easier. When he talks about the intended ...
- 1,465
11
votes
2
answers
721
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Inconsistency and workaday independence.
Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on ...
- 17.6k
12
votes
5
answers
5k
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Proper classes and their consequences
I have two main questions:
What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post ...
- 3,079
21
votes
1
answer
3k
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Philosophical consistency proof for set theory
In his ASL Gödel lecture (Las Vegas, Nevada, 2002), Harvey Friedman asked the following question:
Are there fundamental principles of a general philosophical nature which can be used to give ...
- 261
31
votes
3
answers
5k
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Should there be a true model of set theory?
As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
- 3,982
122
votes
4
answers
39k
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Is the analysis as taught in universities in fact the analysis of definable numbers?
Ten years ago, when I studied in university, I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...
- 10.1k
67
votes
10
answers
14k
views
Arguments against large cardinals
I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...
- 825
18
votes
2
answers
3k
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Universe view vs. Multiverse view of Set Theory
Here I refer to Hamkins' slides:
http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf
particularly, to the "Universe view simulated inside Multiverse", p. 22.
My question is: is it very unsound ...
- 1,465
35
votes
8
answers
4k
views
Interpretation of the Second Incompleteness Theorem
For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:
ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not ...
- 16.2k
12
votes
2
answers
2k
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Proving Independence of Axioms by Exhibiting Models Which Don't Satisfy Our Intuition
I recently saw the proof of the independence of ZF (with allowance for multiple empty sets) and AC. The proof constructed the model based on a set theory generated by infinitely many empty sets and ...
- 15.4k
14
votes
3
answers
2k
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Are there natural examples of mathematical statements which follow from consistency statements?
Motivation
One of the methods for strictly extending a theory $T$ (which is axiomatizable and consistent, and includes enough arithmetic) is adding the sentence expressing the consistency of $T$ ( $...
- 5,502
26
votes
9
answers
8k
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Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]
As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...
9
votes
2
answers
2k
views
Using the multiverse approach to decide the law of the exluded middle?
Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...
user2529
39
votes
5
answers
6k
views
Why do categorical foundationalists want to escape set theory?
This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full.
I know that it's possible to ...
- 855
11
votes
5
answers
9k
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Models of ZFC Set Theory - Getting Started
For just any first-order theory: What are the sets I am supposed/allowed to think of when thinking of models as sets (of something + additional structure)?
Provided:
I can think of models of any ...
- 9,720
40
votes
5
answers
7k
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Were Bourbaki committed to set-theoretical reductionism?
A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent ...
- 1,257
17
votes
8
answers
2k
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The Importance of ZF
It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's ...
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