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The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but rather that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything wrong here, it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but rather that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything wrong here it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but rather that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything wrong here, it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

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The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but rather that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything wrong here it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything wrong here it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but rather that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything wrong here it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

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The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another,. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything "wrong"wrong here it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

This psychological difficultyobstacle is reflected in the long historical struggles with the concept of size for infinite sets, described e.g. in Mancosu's Measuring the Size of Infinite Collections of Natural Numbers numbers: Was Cantor's Theory of Infinite Number Inevitable? It is interesting that Bolzano, Cantor's precursor on set theory, rejected the Hume's principle, exactly because it conflicted with the part-whole axiom. GodelGödel , on the other hand, gave an influential argument that Cantor's choice in favor of the Hume's principle was "inevitable". Nonetheless, a coherent alternative, called numerosity, that adopts the part-whole axiom instead, was introduced by Benci in 1995, and developed by him, Di Nasso and Forti in 2000-s, see their Aristotelian Notion of Size. However, since the Hume's principle is now rejected we can not apply it to sets proper, they need to have additional counting structure, in this case they have to be labeled sets.

The idea is to split a set into boxes, each one containing only finitely many elements (with the same label), and we count them by counting the content of the boxes, in sequence. The numerosity is built from the sequence whose n-th term counts the number of elements with labels up to n (this, i.e. in the first n boxes. This only works for countable sets, but there is an extension to arbitrary sets using ordinals). All integers (labeled by themselves) produce 1,2,3,4,5,6..., even integers (also labeled by themselves) produce 1,1,2,2,3,3..., which is strictly smaller starting from the second term, so their numerosity is strictly smaller. 

The general construction is technical and is similar to construction of the hyperrealultrapower construction of the hyperreal numbers - one takes numerosities to be equivalence classes of nondecreasing integer sequences that are equivalent modulo a Ramsey ultrafilter (whatever this means). But the numerosities do satisfy the part-whole axiom, and have nice algebraic properties. One can define arithmetic on them, and they add when disjoint labeled sets are put together. The trade-off is that there are plenty of bijective sets with different numerosities.

Here is Mancosu's commentary on Godel'sGödel's argument and how it fails for numerosities:

"Godel’sGödel’s reflection aims at showing that in generalizing the notion of number from the finite to the infinite one inevitable ends up with the Cantorian notion of cardinal number. The key step in the argument is the premise and the theory of numerosities can help us see that the premise already contains in itself the Cantorian solution. In fact, the premise takes takes as evident the request that “the number of objects belonging to some class does not change if, leaving the objects the same, one changes in any way whatsoever their properties or mutual relations (e.g., their colors or their distribution in space).” While the premise constitutes constitutes no problem when dealing with finite sets, one might question its acceptability in the realm of the infinite. Indeed, in the theory of numerosities we cannot grant the premise when it comes to infinite sets.

For, while it is possible to abstract from the nature of the objects objects themselves there is one type of relation that affects the counting, namely the way in which the elements are grouped. Such grouping makes no difference in the realm of finite sets of integers. But when we move to infinite sets a rearrangement of the grouping will in general affect the approximating functions and thus the numerosity of the set. Someone committed to the counting embodied in the theory of numerosities might thus reasonably resist accepting the premise on which Godel bases his argument and thus also resist the claim that the generalization of number from the finite to the infinite must perforce end up with the notion of cardinal number. In short, having a different way of counting infinite sets shows that while GodelGödel gives voice to one plausible intuition about how to generalize “number” to infinite sets there are coherent alternatives."

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another, Cantor made one choice, and it was adopted in modern mathematics, but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything "wrong" here it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

This psychological difficulty is reflected in long historical struggles with the concept of size for infinite sets described in Mancosu's Measuring the Size of Infinite Collections of Natural Numbers numbers: Was Cantor's Theory of Infinite Number Inevitable? It is interesting that Bolzano, Cantor's precursor on set theory, rejected the Hume's principle, exactly because it conflicted with the part-whole axiom. Godel, on the other hand, gave an influential argument that Cantor's choice in favor of the Hume's principle was "inevitable". Nonetheless, a coherent alternative, called numerosity, that adopts the part-whole axiom instead, was introduced by Benci in 1995, and developed by him, Di Nasso and Forti in 2000-s, see their Aristotelian Notion of Size. However, since the Hume's principle is now rejected we can not apply it to sets proper, they need to have additional counting structure, in this case they have to be labeled sets.

The idea is to split a set into boxes each one containing only finitely many elements (with the same label), and we count them by counting the content of the boxes, in sequence. The numerosity is built from the sequence whose n-th term counts the number of elements with labels up to n (this only works for countable sets, but there is an extension to arbitrary sets using ordinals). All integers (labeled by themselves) produce 1,2,3,4,5,6..., even integers produce 1,1,2,2,3,3..., which is strictly smaller starting from the second term, so their numerosity is strictly smaller. The general construction is technical and similar to construction of the hyperreal numbers - one takes numerosities to be equivalence classes of nondecreasing integer sequences that are equivalent modulo a Ramsey ultrafilter (whatever this means). But the numerosities do satisfy the part-whole axiom, and have nice algebraic properties. One can define arithmetic on them, and they add when disjoint labeled sets are put together. The trade-off is that there are plenty of bijective sets with different numerosities.

Here is Mancosu's commentary on Godel's argument and how it fails for numerosities:

"Godel’s reflection aims at showing that in generalizing the notion of number from the finite to the infinite one inevitable ends up with the Cantorian notion of cardinal number. The key step in the argument is the premise and the theory of numerosities can help us see that the premise already contains in itself the Cantorian solution. In fact, the premise takes as evident the request that “the number of objects belonging to some class does not change if, leaving the objects the same, one changes in any way whatsoever their properties or mutual relations (e.g., their colors or their distribution in space).” While the premise constitutes no problem when dealing with finite sets, one might question its acceptability in the realm of the infinite. Indeed, in the theory of numerosities we cannot grant the premise when it comes to infinite sets.

For, while it is possible to abstract from the nature of the objects themselves there is one type of relation that affects the counting, namely the way in which the elements are grouped. Such grouping makes no difference in the realm of finite sets of integers. But when we move to infinite sets a rearrangement of the grouping will in general affect the approximating functions and thus the numerosity of the set. Someone committed to the counting embodied in the theory of numerosities might thus reasonably resist accepting the premise on which Godel bases his argument and thus also resist the claim that the generalization of number from the finite to the infinite must perforce end up with the notion of cardinal number. In short, having a different way of counting infinite sets shows that while Godel gives voice to one plausible intuition about how to generalize “number” to infinite sets there are coherent alternatives."

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but that any extension of "size" to infinite sets will be paradoxical, it has to discard one or the other. One can select one group of intuitions to keep, or one can select another. Cantor made one choice, and it was adopted in modern mathematics (for its technical benefits, among other things), but an alternative group of intuitions can be coherently taken instead. We just can not keep it all. If there is anything wrong here it is expecting that our intuitive concept of size, developed from experience with finite sets, would carry over in full beyond the context in which it was developed.

This psychological obstacle is reflected in the long historical struggles with the concept of size for infinite sets, described e.g. in Mancosu's Measuring the Size of Infinite Collections of Natural Numbers numbers: Was Cantor's Theory of Infinite Number Inevitable? It is interesting that Bolzano, Cantor's precursor on set theory, rejected the Hume's principle, exactly because it conflicted with the part-whole axiom. Gödel , on the other hand, gave an influential argument that Cantor's choice in favor of the Hume's principle was "inevitable". Nonetheless, a coherent alternative, called numerosity, that adopts the part-whole axiom instead, was introduced by Benci in 1995, and developed by him, Di Nasso and Forti in 2000-s, see their Aristotelian Notion of Size. However, since the Hume's principle is now rejected we can not apply it to sets proper, they need to have additional counting structure, in this case they have to be labeled sets.

The idea is to split a set into boxes, each one containing only finitely many elements (with the same label), and count them by counting the content of the boxes, in sequence. The numerosity is built from the sequence whose n-th term counts the number of elements with labels up to n, i.e. in the first n boxes. This only works for countable sets, but there is an extension to arbitrary sets using ordinals. All integers (labeled by themselves) produce 1,2,3,4,5,6..., even integers (also labeled by themselves) produce 1,1,2,2,3,3..., which is strictly smaller starting from the second term, so their numerosity is strictly smaller. 

The general construction is technical and is similar to the ultrapower construction of the hyperreal numbers - one takes numerosities to be equivalence classes of nondecreasing integer sequences that are equivalent modulo a Ramsey ultrafilter (whatever this means). But the numerosities do satisfy the part-whole axiom, and have nice algebraic properties. One can define arithmetic on them, and they add when disjoint labeled sets are put together. The trade-off is that there are plenty of bijective sets with different numerosities.

Here is Mancosu's commentary on Gödel's argument and how it fails for numerosities:

"Gödel’s reflection aims at showing that in generalizing the notion of number from the finite to the infinite one inevitable ends up with the Cantorian notion of cardinal number. The key step in the argument is the premise and the theory of numerosities can help us see that the premise already contains in itself the Cantorian solution. In fact, the premise takes as evident the request that “the number of objects belonging to some class does not change if, leaving the objects the same, one changes in any way whatsoever their properties or mutual relations (e.g., their colors or their distribution in space).” While the premise constitutes no problem when dealing with finite sets, one might question its acceptability in the realm of the infinite. Indeed, in the theory of numerosities we cannot grant the premise when it comes to infinite sets.

For, while it is possible to abstract from the nature of the objects themselves there is one type of relation that affects the counting, namely the way in which the elements are grouped. Such grouping makes no difference in the realm of finite sets of integers. But when we move to infinite sets a rearrangement of the grouping will in general affect the approximating functions and thus the numerosity of the set. Someone committed to the counting embodied in the theory of numerosities might thus reasonably resist accepting the premise on which Godel bases his argument and thus also resist the claim that the generalization of number from the finite to the infinite must perforce end up with the notion of cardinal number. In short, having a different way of counting infinite sets shows that while Gödel gives voice to one plausible intuition about how to generalize “number” to infinite sets there are coherent alternatives."

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