All Questions
Tagged with mathematical-philosophy set-theory
117 questions
14
votes
1
answer
2k
views
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 ...
- 3,793
4
votes
2
answers
749
views
What is the impact on Godels theorem of Paraconsistency?
Russells paradox forced a restriction of the natural abstraction principle (that every predicate determines a set) so that Set Theory could be consistent. The standard one being ZF.
However ...
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1
vote
0
answers
134
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A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)
In their paper "The Theory of Sets of Ordinals" (arXiv), Koepke and Koerwien propose a theory SO axiomatizing the class of sets of ordinals in a model of ZFC and show that SO and ZFC are bi-...
- 6,132
1
vote
2
answers
819
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Ontological status of some "sets" in ZFC [closed]
Let $\phi$ be an undecidable statement of ZFC set theory, for example let's take continuum hypothesis.
What is the ontological status of the "set" $X=\bigl\{x\in\{1,2\}:x=1\text{ or }(x=2\text{ and }\...
CommunityBot
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3
votes
0
answers
342
views
A Question Regarding Boolean-valued Models
What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...
CommunityBot
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9
votes
4
answers
1k
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Does there exist a non-trivial Ultrafinitist set theory?
Does there exist a set theory T-which has not yet been proved to be inconsistent-and in which
one can prove the existence of (1) the empty set (2) sets that are singletons and (3) sets which
have non-...
CommunityBot
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1
vote
0
answers
260
views
A question regarding Koepke' s Ordinal Computability in HOD
Consider the following theorem of Koepke-Koerwien-Siders:
"A set x of ordinals is ordinal computable [either by ordinal Turing machines or ordinal register machines--my comment] if and only if it is ...
- 44.4k
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 ...
- 14k
2
votes
1
answer
275
views
comprehension and ideal elements
A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and relations....
- 236k
11
votes
3
answers
2k
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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, ...
CommunityBot
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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 ...
- 666
15
votes
5
answers
2k
views
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 ...
CommunityBot
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11
votes
2
answers
721
views
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 ...
- 2,762
12
votes
2
answers
2k
views
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 ...
CommunityBot
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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 $T$ ( $...
CommunityBot
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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 ...
CommunityBot
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26
votes
9
answers
8k
views
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 ...
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