Injection from Hartogs number to power set
Let $X$ be a set. I need to find an injection from the set of ordinals
$$ Gamma(x) = { alpha : text{there exists an injection $f$ such that} quad f:alpha to X}$$
to $mathcal{P}(mathcal{P}(X times X))$ (without using $AC$).
My reasoning: Since for $alpha in Gamma(X)$ there exists an injection $f : alpha to X$, I am thinking on using $f[alpha] subseteq X$ and somehow build an injection to $mathcal{P}(mathcal{P}(X times X))$ but I am lost. Any hint would be appreciated!
logic set-theory
asked Dec 2 '18 at 6:56
mate89
1799
add a comment |
Let $X$ be a set. I need to find an injection from the set of ordinals
$$ Gamma(x) = { alpha : text{there exists an injection $f$ such that} quad f:alpha to X}$$
to $mathcal{P}(mathcal{P}(X times X))$ (without using $AC$).
My reasoning: Since for $alpha in Gamma(X)$ there exists an injection $f : alpha to X$, I am thinking on using $f[alpha] subseteq X$ and somehow build an injection to $mathcal{P}(mathcal{P}(X times X))$ but I am lost. Any hint would be appreciated!
logic set-theory
asked Dec 2 '18 at 6:56
mate89
1799
add a comment |
Let $X$ be a set. I need to find an injection from the set of ordinals
$$ Gamma(x) = { alpha : text{there exists an injection $f$ such that} quad f:alpha to X}$$
to $mathcal{P}(mathcal{P}(X times X))$ (without using $AC$).
My reasoning: Since for $alpha in Gamma(X)$ there exists an injection $f : alpha to X$, I am thinking on using $f[alpha] subseteq X$ and somehow build an injection to $mathcal{P}(mathcal{P}(X times X))$ but I am lost. Any hint would be appreciated!
logic set-theory
asked Dec 2 '18 at 6:56
mate89
1799
Let $X$ be a set. I need to find an injection from the set of ordinals
$$ Gamma(x) = { alpha : text{there exists an injection $f$ such that} quad f:alpha to X}$$
to $mathcal{P}(mathcal{P}(X times X))$ (without using $AC$).
My reasoning: Since for $alpha in Gamma(X)$ there exists an injection $f : alpha to X$, I am thinking on using $f[alpha] subseteq X$ and somehow build an injection to $mathcal{P}(mathcal{P}(X times X))$ but I am lost. Any hint would be appreciated!
logic set-theory
logic set-theory
asked Dec 2 '18 at 6:56
mate89
1799
asked Dec 2 '18 at 6:56
mate89
1799
asked Dec 2 '18 at 6:56
mate89
1799
asked Dec 2 '18 at 6:56
mate89
1799
asked Dec 2 '18 at 6:56
mate89
1799
1799
add a comment |
add a comment |
1 Answer
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For $alphainGamma(X) ,$ let $g(alpha)$ be the set of all well-orderings of $X$ of type $alpha.$ This function $g$ is injective, and any well-ordering is a subset of $Xtimes X,$ so an element of $mathcal P(Xtimes X),$ so a set of well-orderings of $X$ is an element of $mathcal P(mathcal P(Xtimes X)).$
answered Dec 2 '18 at 7:10
spaceisdarkgreen
32.4k21753
Thanks for your answer sir.
– mate89
Dec 4 '18 at 3:46
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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For $alphainGamma(X) ,$ let $g(alpha)$ be the set of all well-orderings of $X$ of type $alpha.$ This function $g$ is injective, and any well-ordering is a subset of $Xtimes X,$ so an element of $mathcal P(Xtimes X),$ so a set of well-orderings of $X$ is an element of $mathcal P(mathcal P(Xtimes X)).$
answered Dec 2 '18 at 7:10
spaceisdarkgreen
32.4k21753
Thanks for your answer sir.
– mate89
Dec 4 '18 at 3:46
add a comment |
For $alphainGamma(X) ,$ let $g(alpha)$ be the set of all well-orderings of $X$ of type $alpha.$ This function $g$ is injective, and any well-ordering is a subset of $Xtimes X,$ so an element of $mathcal P(Xtimes X),$ so a set of well-orderings of $X$ is an element of $mathcal P(mathcal P(Xtimes X)).$
answered Dec 2 '18 at 7:10
spaceisdarkgreen
32.4k21753
Thanks for your answer sir.
– mate89
Dec 4 '18 at 3:46
add a comment |
For $alphainGamma(X) ,$ let $g(alpha)$ be the set of all well-orderings of $X$ of type $alpha.$ This function $g$ is injective, and any well-ordering is a subset of $Xtimes X,$ so an element of $mathcal P(Xtimes X),$ so a set of well-orderings of $X$ is an element of $mathcal P(mathcal P(Xtimes X)).$
answered Dec 2 '18 at 7:10
spaceisdarkgreen
32.4k21753
For $alphainGamma(X) ,$ let $g(alpha)$ be the set of all well-orderings of $X$ of type $alpha.$ This function $g$ is injective, and any well-ordering is a subset of $Xtimes X,$ so an element of $mathcal P(Xtimes X),$ so a set of well-orderings of $X$ is an element of $mathcal P(mathcal P(Xtimes X)).$
answered Dec 2 '18 at 7:10
spaceisdarkgreen
32.4k21753
answered Dec 2 '18 at 7:10
spaceisdarkgreen
32.4k21753
answered Dec 2 '18 at 7:10
spaceisdarkgreen
32.4k21753
answered Dec 2 '18 at 7:10
spaceisdarkgreen
32.4k21753
32.4k21753
Thanks for your answer sir.
– mate89
Dec 4 '18 at 3:46
add a comment |
Thanks for your answer sir.
– mate89
Dec 4 '18 at 3:46
Thanks for your answer sir.
– mate89
Dec 4 '18 at 3:46
Thanks for your answer sir.
– mate89
Dec 4 '18 at 3:46
add a comment |
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