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## Main Question or Discussion Point

O.k. I am seriously confused... Not being to good at math but nevertheless interested in set theory, infinity, etc. I started reading Mary Tiles, The Philosophy of Set Theory (Dover edition). I particularly wanted to know more about the relation between infinite ordinals and cardinality, but what she writes about that topic is very confusing to me. Let me explain:

Starting from the following two principles Cantor was able to generate the infinite ordinals:

(1) if a is a number, then a+1 is its immediate successor;

(2) if there is a definite succession of numbers of which there is no greatest, then there is a limit or new number which is the next greatest to them all.

Using (1) you get the potential infinity of the natural numbers (0,1,2,3...). According to Tiles, Cantor called this the first number class.

Then using (2) you get the limit of the first number class which is ω, the first infinite ordinal.

Next, starting from ω and applying principles (1) and (2) over and over again, you get ω⋅2, ω⋅3... ω⋅ω... etc. According to Tiles, this is what Cantor called the second number class.

Now if I understand Tiles correctly she says that all the ordinals in the second number class have cardinality alef-0 (see below).

To get to ordinals with higher cardinality Cantor then introduced a third principle the formulation of which by Tiles remains very obscure to me but which seems to me to be an application of principle (2) to the entire succession of ordinals in the second number class. So given the succession ω, ω⋅2, ω⋅3... ω⋅ω... etc. we can (using (2)) create a limit or new number which is the next greatest, namely, ω1.

The same principles can then be appplied to ω1 and the result of this woul be the third number class whose limit is ω2 and so on...

Now comes the rub: if the cardinality of all the ordinals in the second number class is alef-0, the cardinality of the third number class (ω1, ω1+1, etc.) must be alef-1, right?

Yet this is not what Tiles says. She goes on to say some very confusing things. Here are some quotes:

"Cantor proved that the second number class cannot be put into one-one correspondence with the first...." (p.106)

This I find confusing. The first number class is simply the potential infinity of 0,1,2,3...etc. The second number class results simply from considering that potential infinity as an actual one and having its limit in ω. If we use Von Neumann's trick to define a number as the set of the numbers that precede it, then ω={0,1,2,3...etc}. Hence it follows, I would say, that ω can be put in one-one correspondence with {0,1,2,3...etc.} -- in other words, ω is denumerable. But this is not what Tiles says in the quote above.

Second problem... Tiles writes: "Thus the cardinal number of the second number class is the next after [alef-0] and is labeled [alef-1]." (p.107)

But didn't she state earlier that all the ordinals in the second number class have cardinality alef-0? This seems to follow from what she writes earlier, namely:

"All the numbers in the second number class can, however, be thought of as the numbers obtained by introducing more or less complicated order on the sequence of natural numbers... This means that although we have generated a lot of infinite ordinal numbers [...] they are all such that they are ordinal numbers of sets which can be put in one-one correspondence with the the natural numbers (denumerable sets)... So what we have is a proliferation of infinite ordinal numbers which all apply to sets having the same cardinality, [alef-0]. These first two principles [namely, (1) and (2)] on their own do not generate any ordinal number which could be the number of points in a line [i.e. alef-1]."

According to me, Tiles is very confused here. On the one hand she says that principles (1) and (2) generate the second number class and this has cardinality alef-0. But she also says that the cardinality of the second number class is alef-1!

Am I mistaken, or is Tiles contradicting herself here. Please enlighten me!

Starting from the following two principles Cantor was able to generate the infinite ordinals:

(1) if a is a number, then a+1 is its immediate successor;

(2) if there is a definite succession of numbers of which there is no greatest, then there is a limit or new number which is the next greatest to them all.

Using (1) you get the potential infinity of the natural numbers (0,1,2,3...). According to Tiles, Cantor called this the first number class.

Then using (2) you get the limit of the first number class which is ω, the first infinite ordinal.

Next, starting from ω and applying principles (1) and (2) over and over again, you get ω⋅2, ω⋅3... ω⋅ω... etc. According to Tiles, this is what Cantor called the second number class.

Now if I understand Tiles correctly she says that all the ordinals in the second number class have cardinality alef-0 (see below).

To get to ordinals with higher cardinality Cantor then introduced a third principle the formulation of which by Tiles remains very obscure to me but which seems to me to be an application of principle (2) to the entire succession of ordinals in the second number class. So given the succession ω, ω⋅2, ω⋅3... ω⋅ω... etc. we can (using (2)) create a limit or new number which is the next greatest, namely, ω1.

The same principles can then be appplied to ω1 and the result of this woul be the third number class whose limit is ω2 and so on...

Now comes the rub: if the cardinality of all the ordinals in the second number class is alef-0, the cardinality of the third number class (ω1, ω1+1, etc.) must be alef-1, right?

Yet this is not what Tiles says. She goes on to say some very confusing things. Here are some quotes:

"Cantor proved that the second number class cannot be put into one-one correspondence with the first...." (p.106)

This I find confusing. The first number class is simply the potential infinity of 0,1,2,3...etc. The second number class results simply from considering that potential infinity as an actual one and having its limit in ω. If we use Von Neumann's trick to define a number as the set of the numbers that precede it, then ω={0,1,2,3...etc}. Hence it follows, I would say, that ω can be put in one-one correspondence with {0,1,2,3...etc.} -- in other words, ω is denumerable. But this is not what Tiles says in the quote above.

Second problem... Tiles writes: "Thus the cardinal number of the second number class is the next after [alef-0] and is labeled [alef-1]." (p.107)

But didn't she state earlier that all the ordinals in the second number class have cardinality alef-0? This seems to follow from what she writes earlier, namely:

"All the numbers in the second number class can, however, be thought of as the numbers obtained by introducing more or less complicated order on the sequence of natural numbers... This means that although we have generated a lot of infinite ordinal numbers [...] they are all such that they are ordinal numbers of sets which can be put in one-one correspondence with the the natural numbers (denumerable sets)... So what we have is a proliferation of infinite ordinal numbers which all apply to sets having the same cardinality, [alef-0]. These first two principles [namely, (1) and (2)] on their own do not generate any ordinal number which could be the number of points in a line [i.e. alef-1]."

According to me, Tiles is very confused here. On the one hand she says that principles (1) and (2) generate the second number class and this has cardinality alef-0. But she also says that the cardinality of the second number class is alef-1!

Am I mistaken, or is Tiles contradicting herself here. Please enlighten me!