Question about Generalized Continuum Hypothesis [closed]

I wonder how the Generalized Continuum Hypothesis reveal that A×A is equivalent to A?
A is any infinite set.

Answer

It is relatively easy to see the outline of the proof (over ZF) that GCH implies A×AA for all infinite sets A. This is usually presented as an intermediate step towards the meatier result that GCH gives us choice. Let me give a sketch.

Assume GCH. Note that if m is an infinite cardinality, and there are no intermediate sizes between m and 2m, then m+1=m. This is because one can see directly that m+1<2m for all m>1 (generalizing slightly Cantor's argument for 2m>m).

But then we have that m+m=m, because mm+m2m+2m=2m+1=2m. This is because 2m<2m. (In fact, over ZF, we have nm<2m for all finite n. This is a result of Specker. A stronger result is that if 0 injects into X, then P(X) cannot inject into the set of finite sequences of elements of X. This was proved by Halbeisen and Shelah, see for example this blog post of mine.)

Finally, m2=m, since mm2(2m)2=22m=2m, and 2m (by the Halbeisen-Shelah result, for example.)

Using the Halbeisen-Shelah result here is an overkill, of course. Specker's original argument established 2^{\mathfrak m}\not\le \mathfrak m^2 directly from the assumption \mathfrak m\ge5.


Note that the argument above is "local" in the sense that it concludes \mathfrak m^2=\mathfrak m from the sole assumption of \mathsf{GCH} at \mathfrak m. Since the assumption that \mathfrak m^2=\mathfrak m for all \mathfrak m implies choice, the ideal result here would be to show that from the \mathsf{GCH} at \mathfrak m it follows that \mathfrak m is well-orderable (that is, an aleph). Sierpiński proved that the well-orderability of \mathfrak m follows from \mathsf{GCH} at \mathfrak m, 2^{\mathfrak m}, and 2^{2^{\mathfrak m}}. Specker improved this, showing that assuming \mathsf{GCH} at \mathfrak m and 2^{\mathfrak m} suffices. Whether \mathsf{GCH} at \mathfrak m is already enough is an open problem.

Attribution
Source : Link , Question Author : user64030 , Answer Author : Andrés E. Caicedo

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