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Results tagged with large-cardinals
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user 82843
31
votes
8
answers
3k
views
On independence and large cardinal strength of physical statements
The present post is intended to tackle the possible interactions of two bizarre realms of extremely large and extremely small creatures, namely large cardinals and quantum physics.
Maybe after all tho …
19
votes
2
answers
8k
views
The Ultimate L in a Nutshell: On Descriptive Articles
Everybody who catches a fleeting glimpse of Woodin's central papers on Ultimate $L$ (i.e. Suitable Extender Models I & II), admits that they aren't so tempting for lazy readers who don't like to deal …
13
votes
3
answers
1k
views
Cops, Robbers and Cardinals: The Infinite Manhunt
Cops & Robbers is a certain pursuit-evasion game between two players, Alice and Bob. Alice is in charge of the Justice Bureau, which controls one or more law enforcement officers, the cops. Bob contro …
9
votes
1
answer
595
views
On the Large Cardinal Strength of Normal Moore Space Conjecture
In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces:
Theorem. (Jones) If $2^{\aleph_0}<2^{\aleph_1}$ then all separable normal Moore spaces are metrizable.
Then h …
8
votes
2
answers
308
views
Does fast function forcing really have $\kappa$-Knaster property?
I ran into a claim concerning Woodin's fast function forcing in the following paper of Apter and Cummings which sounds no right to me:
A. Apter, J. Cummings, Blowing up the power set of the least meas …
7
votes
2
answers
761
views
Which large cardinals have a Matryoshka characterization?
What on Earth do Russian Matryoshka dolls have in common with large cardinal axioms?! Well, the answer lies in Jónsson algebras! Here is how:
As illustrated in the pictures, a Matryoshka set is a self …
5
votes
2
answers
533
views
Statements that Could be Forced by Ultrapowers
Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good propertie …
5
votes
1
answer
399
views
Can Woodin's fast function forcing kill Shelah cardinals?
Definition 1. An uncountable cardinal $\kappa$ is Shelah if for every function $f:\kappa\rightarrow \kappa$ there exists a transitive class $M$ and a non-trivial elementary embedding $j:V\rightarrow M …
5
votes
1
answer
673
views
What are examples of non-equivalent virtualizations of a large cardinal?
This is a follow up to my previous question concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called virtualization …
3
votes
1
answer
328
views
What is the consistency strength of non-existence of outer automorphisms of Calkin algebra?
The Calkin algebra $C(H)$ is the quotient of $B(H)$, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space $H$, by the ideal $K(H)$ of compact operators.
In 1977, Br …
3
votes
1
answer
764
views
Is tree property inconsistent with Berkeley cardinals in the absence of Axiom of Choice?
On one hand due to Kunen's inconsistency theorem it is known that within $\sf ZF$, large cardinal axioms beyond Reinhardt cardinal are inconsistent with $\sf AC$.
Also some recent results of Bagaria, …
3
votes
1
answer
442
views
A Question on HOD, V and GCH
The theorem 1.1 of the following paper:
Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic.
states that:
Theorem: Assume $V\models ZFC+GCH+~\text{There exists a}~(\kappa+4)-\text{str …
2
votes
1
answer
636
views
On the Actual Potential of Virtual Large Cardinals
Virtual large cardinals belong to a relatively new breed of strong axioms of infinity. They often appear as statements of the following form:
Definition. Suppose $A$ is a large cardinal property c …
2
votes
1
answer
385
views
Strong Total Failures vs. Weak Instances of the Generalized Continuum Hypothesis
The exponentiation operator inflicts a subtle information loss on the transfinite numerical equations, pretty similar to the case of $a^2=b^2 \nRightarrow a=b$ in real numbers. In fact, for the infini …