# Tag Info

### Ordering of large cardinals by cardinality

To elaborate on Joel David Hamkins's answer: When the the size order of large cardinal properties differs from the strength order and it is not an example of the identity crisis phenomenon, it is ...
Accepted

Accepted

### A continuous map relating co-constructible reals

By "formula" I will mean "formula of set theory with ordinal parameters." Note that "ordinal parameters" is equivalent to "constructible parameters" for our ...
• 21.5k

### Can countable ordinals start gaps of every order in the constructible universe?

Your other questions have been answered, so I provide here only an answer for your last edit, that is, whether an ordinal can be in a gap of every order. The answer is positive by an argument almost ...
• 361
1 vote
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### Consistency strength of an attempt at higher order set theory

I believe $ZFC$ plus the existence of a countable collection of strictly inaccessible cardinals is also a lower bound on the consistency strength of this theory, and am posting this as a CW answer to ...
Accepted

### Optimal partitions amongst a given set of partitions

Modifying your note at the end slightly, try the following. Let $X=\{0,1,2,\ldots\}$ and let $P=\{\{0,1\},\{2,3\},\{4,5\},\ldots\}$. Let $\mathfrak{P}$ be all partitions of $X$ which refine $P$ and ...
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• 395
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### Given $\pi$ permutation on $\{1,\dotsc,n\}$, what is the sign of a permutation of $\{2,\dotsc,\hat\jmath,\dotsc,n\}$?

$\DeclareMathOperator\sgn{sgn}$This is almost the same as your previous question, just with the order of the operations switched—whether you think of $\pi$ as ordering or disordering is just a matter ...
• 7,933
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### Can a stage of the cumulative hierarchy violate the partition principle?

If you are asking whether or not a $V_\alpha$ could violate the partition principle, the answer is easily yes. As we all know, it is always the case that $\Bbb R$ can be partitioned into $\aleph_1$ ...
• 35.4k
Accepted

### Sign of the permutation which brings a subsequence back to its original form

Imagine a bubble sort where you bring each element to its original position. $x_1$ would have taken $\pi^{-1}(1) - 1$ transpositions to bring it back to its original position. Let's perform those ...
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• 21.5k
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### Product topology from two premetric spaces induced by sum of premetrics?

The answer to this question is negative. Consider the subspace $M_1=\{0\}\cup\{\frac 1n+\tfrac{i}{nm}:n,m\in\mathbb N\}$ of the complex plane and the space $M_2=M_1\cup\{\frac1n:n\in\mathbb N\}$ ...
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### Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA?
I think that the answer to both of your questions is negative. Let $X$ be a subspace of the real line with its usual topology of size $\omega_1$. Then, clearly, $X$ is first countable, Lindelöf and ...