Talk:Galileo's paradox

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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)