Talk:Galileo's paradox

Opening comments

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)

Ahhh! I think I get it now. It's talking about a finite fraction of an infinite set still being infinite. I still think the sentence is misleading, though, but I'm not sure how to fix it.--Andrew Eisenberg 22:57, 8 August 2006 (UTC)

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)

________________ While I am not confused, I do not accept obvious attempts of deception.

i) "This is an early use, though not the first, of a proof by one-to-one correspondence of infinite sets." Is it really a proof? A proof of what? I see it just an early case of infinite one-to-one correspondence. Already Albert of Saxony (1316-1390) came in his book "Questiones subtilissime in libros de celo et mundi" to the same conclusion that an wooden bar has as many points as the whole three-dimensional space.

ii) "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."

The words 'applied' and 'did' are misleading. What Salvati understood is still true. The sentence is less clear as compared with the final conclusion by Salvati, representing Galilei himself: "... the attributes =,>, and <, are not applicable to infinite, but only to finite, quantities."

Moreover, a quantity is not a set. I do not support the definition by v. Helmholtz: "Objects are quantities if they allow the relations =,>,<" but I recommend to clarify the meaning of set. Notice: According to Fraenkel 1923, Cantor's definition of a set has been declared invalid with nor substitute definition.

iii) "Cantor, using the same methods, ..."

Cantor merely used one-to-one correspondence, too.

iv) "... showed that while Galileo's result was correct as applied to the whole numbers and even the rational numbers, the general conclusion did not follow:..."

He merely claimed this. His second diagonal argument merely shows that real numbers are uncountable while Cantor called them more than countable. I do not see any reason to restrict Salvati's conculsion. On the contrary, it furnishs the correct, maybe boring, basis of mathematics.

v) "... some infinite sets are larger than others, in that they cannot be put into one-to-one correspondence."

This is no evidence for a larger size. It also applies for the relationsip between the infinite natural numbers and a finite part of them. Blumschein 16:14, 6 March 2007 (UTC)

Further reading

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)

Resolution

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)

Postulation

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)

(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)

Another problem

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)

Errm, no William M. Connolley (talk) 21:02, 12 November 2013 (UTC)

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)

Rm: the number of points in a line segment is the same as the number in a larger line segment: why

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)