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Results tagged with set-theory
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user 5903
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.
2
votes
A Bijection Between the Reals and Infinite Binary Strings
How about Cantor's own argument as on page 488 of Part 1 of his Beiträge?
- 11.4k
7
votes
How to construct a continuous finite additive measure on the natural numbers
Eric van Douwen constructed a whole slew of finitely additive measures on the set of natural numbers, with various shifting and scaling properties in this paper. He discussed how much `choice' was nee …
- 11.4k
2
votes
Axiomatic Set Theory
And if you happen to be (able to read) Czech try Sochor's Metamatematika Teorii Mnozin
9
votes
"Mächtigkeit" versus "Kardinalität"?
Here is Cantor's Beiträge zur Begründung der transfiniten Mengenlehre (Erster Artikel). Read the bottom four lines on the first page: ",Mächtigkeit' oder ,Cardinalzahl' von $M$ nennen wir $\ldots$". T …
- 11.4k
3
votes
Accepted
Injective choice function for non-separable $T_2$-spaces
Here is a partial answer: if $|X|^{\aleph_0}=|X|$ then the answer is yes.
First take an injective function $F:[X\times X]^\omega\to X$ and then take some
function $G:[X\times X]^\omega\to X$ such that …
- 11.4k
9
votes
Products of Baire spaces
There are even two normed Baire spaces whose product is not Baire. See J. van Mill and R Pol, The Baire category theorem in products of linear spaces and topological groupsn, Topology Appl. 22 (1986) …
- 11.4k
6
votes
$|\mathsf{RO}(X)|$ vs. $|\tau_X|$ for Tychonoff spaces
An answer to the first question: the Niemytzki plane, $N$, is an example. Note first that, in general, if $U$ and $V$ are regular open and $D$ is dense then $U=V$ iff $U\cap D=V\cap D$. From this we f …
- 11.4k
11
votes
4
answers
2k
views
Earliest diagonal proof of the uncountability of the reals.
I cited the diagonal proof of the uncountability of the reals as an example of a `common false belief' in mathematics, not because there is anything wrong with the proof but because it is commonly bel …
- 11.4k
6
votes
Who was the first to propose a formal definition of infinity?
In the preface of the second edition of Was sind und was sollen die Zahlen Dedekind mentions another definition of `finite': a set $S$ is called finite if there is a map $\varphi$ from $S$ to itself s …
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12
votes
Independent families of subsets of $\mathbb N$ of size continuum
Let $2^{<\omega}$ be the binary tree and assign to each branch $x$ the family $F_x$ of finite sets that intersect it. If $x_1$, $x_2$, $\ldots$ $x_k$ is a finite set of (distinct) branches then there …
- 11.4k
10
votes
Accepted
What is the extent of a $\Sigma$-product of a (uncountable) power of a (countable) discrete ...
To answer the explicit question: the extent of every $\Sigma$-product of $\mathbb{N}$ is countable. H. H. Corson showed in Normality in subsets of product spaces, Amer. J. Math 81(1959), 785–796 that …
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9
votes
Accepted
A strictly descending chain of subalgebras of $P(\omega)/_{\mathrm{fin}}$
There is a family $\{K_X:X\subseteq\mathfrak{c}\}$ of separable compact
zero-dimensional spaces such that there is a continuous surjection of $K_X$
onto $K_Y$ if and only if $X\subseteq Y$.
These spac …
- 11.4k
3
votes
Questions on continuum hypothesis
As the comment says: the roles of Gödel and Cohen are reversed.
One cannot disprove CH using ZF(C), so I take it you simply assume the negation of CH and ask about fields between $\mathbb{Q}$ and $\m …
- 11.4k
1
vote
About the existence of a particular kind of "splitting" function on atomless complete Boolea...
I think you are asking too much.
Assume we have such a function and let $a$ be nonzero such that both $a_0$ and $a_1$ are nonzero. Then $b\le a_0$ implies $b_1=0$ and $b\le a_1$ implies $b_0=0$.
If …
- 11.4k
2
votes
Countable support product of Sacks forcings and selective ultrafilters
I'm not allowed to comment yet but your question looks equivalent to having the set $A$ in $\mathrm{HL}_\omega$ (see Laver's paper) belong to the ultrafilter. I do not recall how flexible Laver's proo …
- 11.4k