All Questions
6
questions
3
votes
1
answer
116
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In ZF, is it possible that there is no cardinal such that Reals injects into?
Working in ZF, is it possible that there is no cardinal number such that $\mathbb{R}$ can inject into? For if there exists a cardinal number $\kappa$ such that $\mathbb{R}$ injects into $\kappa$, then ...
- 782
3
votes
0
answers
74
views
How high in the constructible hierarchy do you need to go to see Dedekind-incompleteness?
This is a follow-up to my questions here and here. Let $X= (A,+,*,<)$ be an ordered field. Let us define a constructible hierarchy relative to $X$ as follows. Let $D_0(X)=A\cup A^2 \cup \{+,*,&...
- 10.3k
5
votes
0
answers
138
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Where is the first gap in the constructible hierarchy relative to a real closed field?
This is a follow-up to my question here. Let $X= (A,+,*)$ be a real closed field. Let us define a constructible hierarchy relative to $X$ as follows. Let $D_0(X)=A\cup A^2 \cup \{+,*\}$. For any ...
- 10.3k
2
votes
1
answer
105
views
Where is the copy of $\mathbb{N}$ in the constructible hierarchy relative to a real closed field?
Let $X$ be a real closed field. Let us define a constructible hierarchy relative to $X$ is defined as follows. (This is slightly nonstandard terminology.). Let $L_0(X)=X$. For any ordinal $\beta$, ...
- 10.3k
1
vote
2
answers
422
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Real numbers for beginners
I am thinking about the Wikipedia (I understand disputed) article about “definable real numbers”.
It begins to say that,
A real number $a$ is first-order definable in the language of set theory, ...
- 1,817
3
votes
1
answer
1k
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Undefinable Real Numbers
Disclamer: I'm sure my definition of "definable" may be different than the/a established mathematical one, I am more than interested in learning why/how this is so, but that is not my question
Part 1:...
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