# Tagged Questions

107 views

### The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
200 views

### Uncountable Cardinals without AC

I am doing an exercise, proving that without AC or Replacement that there are uncountable cardinals. As a point of reference I looked at the proof in Kunen's "The Foundations of Mathematics" that ...
139 views

### Understanding Zorn's lemma.

A lot of authors assume Zorn's lemma. I am told it is not an obvious mathematical fact, but I am having problems understanding why that is. Zorn's lemma states that if every chain in a partially ...
68 views

### Axiom of choice and an example of a Well-ordered $\Bbb R$

From the axiom of choice we get that every set can be ordered in a way that will make it a well ordered set, including $\Bbb R$. However, since the ordinal of such a well-ordered set of $\Bbb R$ will ...
53 views

### A well ordering on $\mathbb{R}$ and bigger sets

Consider the set of sequences $S = \{f:\mathbb{N}\to\mathbb{N}\}$, define an order on $S$ by the following: Based on the well-ordering of $\mathbb{N}$ and induction, either $f_1 = f_2$ or there is a ...
47 views

### On Counted Languages

In my recent question on Godel Completeness I mentioned that there was a related question I wanted to ask, but would keep separate. I have been recently studying "non-well ordered sets" and Chapter 7 ...
45 views

86 views

### Existence of surjection implies existence of injection? [duplicate]

Let $A$ and $B$ be sets. If there exists a surjection $f : A \to B$ then there exists an injection $g : B \to A$. Proof: given $b \in B$ select an element $a \in f^{-1}(b)$. Denote this element by ...
70 views

103 views

### Why do sequences exist? What does “constructing a sequence” mean formally?

Everybody knows arguments like: "We can construct such a sequence inductively. Let $a_0$ be chosen as [..]. Then we can choose $a_{k+1}$ out of the set $A_{k+1}$ (which was shown to be non-empty)." ...
99 views

### AC iff $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$

I'm trying to do the following exercise: Exercise. Assume ZF. Then AC is equivalent to the statement that $P(\delta)$ can be well-ordered for all $\delta\in{\bf On}$. I'm struggling with the ...
93 views

151 views

### If $B\not\prec A$, does $A\preceq {\cal P}(B)$ or $A\preceq {\cal P}{\cal P}(B)$, etc in ${\sf ZF}$?

I've heard it said that theorems based on choice are also available in ZF "a few powersets away", and I think this is one of them, but I'm not sure how to prove it. (I'm also interested to hear of ...
141 views

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ... 1answer 115 views ### Is it possible to prove$|V_\kappa|=\kappa$for strongly inaccessible$\kappa$without AC? First, let me note that my definition of "strongly inaccessible" is that a nonzero ordinal$\kappa$is strongly inaccessible if${\rm cf}(\kappa)=\kappa$and$\forall\alpha<\kappa, {\cal ...
Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ...