Talk:Galileo's paradox

message: I'm confused:
 Galileo concluded that the ideas of less, equal, and 
 greater applied only to finite sets, and did not make 
 sense when applied to infinite sets.
Even in the infinite set of positive numbers, we know 3 < 4 and 4 > 3 and 4 = 4.  Is this talking about the size of sub-sets?  I'm not a mathematician, so I may just be misunderstanding the meanings of the terms less, greater and equal.--Andrew Eisenberg 22:50, 8 August 2006 (UTC)
replies: text: It seems quite clear to me, and correctly phrased.  It is not saying anything about applying those operations to elements of finite or infinite sets, as you did above, but to the sets themselves.  I  think you should leave it alone.  -- Dominus 02:38, 9 August 2006 (UTC)
thread_title:  Opening comments 

message: Can anyone recommend a source for this? It would be nice to see it phrased in terms of subsets of N (using straight mathematical terminology). 118.90.51.217 (talk) 04:50, 4 November 2008 (UTC)
thread_title:  Further reading 

message: Telling from some comments in the discussion I thought that some people might appreciate a "mathematical" resolution of the paradox. I hope that this thought is correct and doesn't in any way bother someone (especially people with a higher mathematical education for whom this should be a triviality). Unfortunately, I'm not a native English speaker and trust that someone will correct my linguistic mistakes in the article :). --TheLaeg (talk) 16:10, 17 May 2010 (UTC)
thread_title:  Resolution 

message: But, you can also have the square of a number that is itself a square- For example 2 squared is 4, and 4 squared is 16. then you have 1 non-square and 2 squares, so there must be more squares than non-squares! Right? I can't be the first person to think of this, right? 74.96.158.81 (talk) 12:24, 27 July 2010 (UTC)
replies: text: (Three years later…) Indeed. For all (I think it is all) pairs of infinite sets of the same cardinality, you can make a surjection of either set into the other, or to put that another way, you can argue for there being "more" of it than the other by using a many-to-one correspondence. The fact that equally valid arguments can be made both ways means that the sets are infinite, and equal. Other examples include:
* Every integer N corresponds to two even numbers, 4N-2 and 4N. So 1 maps to 2 and 4, while 2 maps to 6 and 8, and 3 maps to 10 and 12, etc. This schema covers all the even numbers, and it (naively) suggests that there are more evens than integers. Of course, the same could be done the other way around, mapping each even number E with itself and E-1.
* Every integer that doesn't start with a string of consecutive 5s maps onto an infinite number of integers that do. So 1 maps to {51, 551, 5551, etc} while 780 maps onto {5780, 55780, etc}. This suggests that there are infinitely many more numbers that do start with 5s than numbers that don't. (Notice the parallel between this and the intutive but false observation that there must be infinitely many more rationals than integers.) ± Lenoxus (" *** ") 23:42, 16 August 2013 (UTC)
thread_title:  Postulation 

message: A serious mistake:
 [...] and that by this definition some infinite sets are strictly larger than others.
That some infinite sets are strictly larger than others is a consequence of the diagonal argument or equivalent, not a direct consequence of the definition of cardinality.
LudovicoVan (talk) 00:24, 13 July 2012 (UTC)
replies: text: Errm, no William M. Connolley (talk) 21:02, 12 November 2013 (UTC)
replies: text:  Diagonalization is a proof method that can be used to show that there exist infinite sets of differing cardinality. But such are the innovations of Cantor, not Galileo.  — Preceding unsigned comment added by Mmpozulp (talk • contribs) 19:14, 6 December 2014 (UTC)
thread_title:  Another problem 

message: I removed:
: It is nonetheless remarkable the extent by which Galileo anticipated later work on infinite numbers. He showed that the number of points in a line segment is the same as the number in a larger line segment, but of course he didn't discover Cantor's proof that this is greater than the number of integers. 
because its wrong. Galileo came to the conclusion that infinities can't be compared. Therefore, he can't have concluded that "the number of points in a line segment is the same as the number in a larger line segment" William M. Connolley (talk) 21:04, 12 November 2013 (UTC)
thread_title:  Rm: the number of points in a line segment is the same as the number in a larger line segment: why