On Monday, September 16, 2024 at 9:12:58 PM UTC-6 John Clark wrote:
On Mon, Sep 16, 2024 at 9:36 PM Alan Grayson <[email protected]> wrote: > How would you map (0,1) 1-1 onto the real numbers? *F(x)=1/2 + 1/π Arctan(x) . The domain is all the real numbers and the range is (0.1)* *This map isn't 1-1. Many x's correspond to the same point in (0,1). AG * *> **Both sets have the same cardinality.* * And a short line in a long line have the same cardinality, and there is a very simple geometric proof showing that they can be put into a one to one correspondence. Just draw a triangle with a line from the apex running through the short line to the long line at the base of the triangle * *> If a set is infinitely large then there is a proper subset of that set that can be put into a one to one correspondence with the entire set, in fact that is the mathematical definition of "infinity".* *> That's circular. AG * *I deleted the post with the circular comment. Why are you responding to it? AG * *Alan, if you open any book on set theory you will find that the formal mathematical definition of infinity is that something is infinite if and only if a proper subset of it can be put into a one to one correspondence with the entire thing. * *Returning to the focus of this thread is my claim that an infinite universe cannot be created. If one exists, it is eternal. The reason is because the creation would require something non-physical; infinite spatial expansion instantaneously. So no BB for a spatially infinite universe. The situation for a finite universe is different. It can start small and expand, notwithstanding the fact that we don't know how any universe can be created. AG * -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/274a97a9-c5ca-4138-9ca0-7f4e2d1936e1n%40googlegroups.com.
