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Results tagged with set-theory
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user 3106
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
4
votes
Plausibility argument for a measurable cardinal
The standard discussion of the justification of measurable cardinals is Penelope Maddy's article Believing the axioms. Regarding the "unreachability from below" argument, which she calls inexhaustibi …
- 82.6k
5
votes
When do we get $CON(ZF)$ in transfinite progressions of consistency statements?
There is no known explicit combinatorial description of the proof-theoretic ordinal of ZFC. Even much weaker set theories have so far defied explicit description. For a recent account that gives some …
- 82.6k
13
votes
What are the differences between Woodin and Sy Friedman regarding set theoretic truths?
It is only a half-joke to say that for a layman, there is no difference between their approaches. Take the following question for example: Does there exist an uncountable subset of $\mathbb R$ that c …
- 82.6k
15
votes
Circular, or missing, definition in set theory?
I'm not sure I understand your question, since at first it sounds like you're thinking of $\in$ as a multi-valued function that sends a set $A$ to an element $x$ of $A$, but then I would expect you to …
- 82.6k
6
votes
Axiom of choice in linear algebra
Your question is essentially answered in a math.SE post by Asaf Karagila, but I think it is worth spelling out a subtle point. The axiom of multiple choice (or MC for short) says:
For every set $X$ …
9
votes
reducing a theorem to set theory using first order logic
Deducing a non-trivial theorem directly from ZFC is a tedious business. First you will need to define the integers in terms of sets. The natural numbers are most commonly encoded as von Neumann ordi …
- 82.6k
15
votes
Why is an internal proof of consistency satisfactory for some systems?
The answer by user57888 is correct, but let me emphasize two things. The first is that much of the interest in this type of question predates Gödel's theorems. So if you want to understand the origina …
- 82.6k
17
votes
What axioms are used to prove Gödel's Incompleteness Theorems?
You might already know this, but if you're looking for foundations of mathematics which are so weak that they don't prove the existence of non-r.e. sets, then you should study Simpson's book Subsystem …
- 82.6k
1
vote
What is the idea behind stationary sets?
This answer overlaps with other answers, but I think it’s worth mentioning separately because I find, as a general principle, that one good way to motivate a concept is to see how it can be used to so …
- 82.6k
10
votes
Simple bijection between reals and sets of natural numbers
I think this question is more interesting than it appears at first glance.
The answer depends partly on how you define a real number. For example, a standard way to define real numbers is by means o …
- 82.6k
5
votes
Forcing as a tool to prove theorems
Matthew Wiener once explained to me that because of the close connections between forcing and the Baire category theorem, forcing could be used to prove certain results in analysis that are more commo …
- 82.6k
4
votes
Parts of Set Theory immune to independence
I'm not sure if this is the kind of thing you're looking for, but the assumption $V=L$ settles most "interesting" questions in set theory. For example, it implies the generalized continuum hypothesis …
- 82.6k
18
votes
2
answers
1k
views
Must uncountable standard models of ZFC satisfy CH?
In Cohen's article, The Discovery of Forcing, he says that "one cannot prove the
existence of any uncountable standard model in which AC holds, and
CH is false,"
and offers the following proof.
If $M …
- 82.6k
57
votes
Can we prove set theory is consistent?
Your question certainly makes sense and it is a point that I feel is too often glossed over in textbooks.
Let me rephrase your question. Goedel's second theorem says that, assuming that a certain fo …
- 82.6k
13
votes
Accepted
Failure of the GCH
The following quotations are taken from Matthew Foreman and W. Hugh Woodin, "The generalized continuum hypothesis can fail everywhere," Ann. Math. 133 (1991), 1–35.
THEOREM. Let $\kappa$ be a superc …
- 82.6k