3. The properties and relations of any set of physical objects may be changed arbitrarily.

This is an interesting condition to consider because without it I’d argue that the Euler number satisfies many of the properties you’d expect of a notion of ‘size’, surprisingly many in fact.

]]>That is to say, Z and 2Z being the same as sets just means that we must have forgotten the fixed inclusion between them when we say “as sets.” In fact, one could take this as a proof that one cannot reconstruct the inclusion after forgetting it (as opposed to, say, the fixed map from Z to a ring).

And my point was just that not paying attention to what you’ve remembered and what you’ve forgotten is very dangerous.

]]>Just as there are times that we make progress by forgetting structure, I think there are times that we make progress by realizing that some level of structure is actually important. I think this is why complex analysis for instance often says much mor of interest than real analysis, because there’s an extra level of structure implied by differentiability that makes things manageable.

]]>The art of mathematics is the art of *cleverly forgetting structure*. Thus any time you want to define a notion it is crucial to fix which structure you are remembering and which structure you are forgetting. Intuitions that are inherited from forgotten structure *must be ignored*. It is only by forgetting structure that we are able to generalize and make progress.

In this example, clearly it makes sense to argue using this intuition that the even integers are smaller than the integers if we are remembering their structure as *subsets of the integers*. However, if we have explicitly forgotten that and are only remembering their set structure (which is the entire point of *set theory*), then it is total nonsense to make that claim. And worse, pernicious nonsense that attacks the fundamental idea of mathematics.

I’d like to hear more about your suggestion that our apparently conflicting intuitions about size are in fact consistent, since they track different notions. If that’s right, then we have a solution to Galileo’s paradox that doesn’t require discrediting any of the intuitions, which would be ideal from my point of view.

I wonder how plausible it is, though. Consider Kripke’s claim that there are two notions possibility, and that our intuition that it’s possible that water contains no hydrogen is consistent with our intuition that it’s impossible for there to be water that contains no hyrogen. Once the difference is pointed out, we can really hear the two different readings. We see, clearly, that there is a sense in which it is possible and a sense in which it’s not. By contrast, when we ask whether there are as many evens as naturals, we don’t hear two readings. I feel no comparable temptation to say: there’s a sense in which there are as many evens as naturals and a sense in which there aren’t. A good comparison might be Newcomb’s paradox. We feel the pull of saying it’s rational to choose the one box, and we feel the pull of saying it’s rational to choose both boxes. But do we really have two separate notions of rationality?

I don’t mean to deny that ‘as many as’ has one meaning in the mouths of set theorists and another in the mouths of the folk. That may be right; perhaps the set theorist just *means* sameness of cardinality (i.e., he hasn’t discovered, so much as stipulated, a definition of equinumerosity). But, even if that’s right, that doesn’t take us any way towards resolving the paradox: while the set theorist’s notion may be consistent, the folk notion still threatens to lead to paradox.

]]>I take the point I made about the evens and the evens minus 2 being intuitively the same size as the odds to indicate that there is no way to extend a fine grained notion of size (until every set has a size) in a way that respects our intuitions. My instinct is to say that it is an analytic truth that sizes should be totally ordered, so the notion you’re defining just doesn’t seem like a notion of size to me.

You mentioned that this is a problem for the bijective conception of size too if we remove the WO property. But similarly removing extensionality puts a spanner in the works for the principle about proper subsets being smaller (ok, you might laugh at the comparison between extensionality and choice, but it feels like choice functions are as essential to the notion of 1-1 correlation as extensionality is to subset relations, and it would be unfair to remove either.) Even if we reject choice we have a fruitful notion of size for the well-orderable sets, whereas the proposed notion doesn’t seem to fair so well, even on the countable sets.

]]>There is supposed to be a conflict of intuitions between (i) the claim that the size of a set is larger than the size of its proper subsets and (ii) the claim that the size of two sets is the same if they can be put into 1-1 correspondence.

The conclusion we are supposed to draw is that the intuitive notion of number is as underspecified as the intuitive notion of set allegedly is. But why are we “blaming” the intuitive notion of number for this paradox rather than the intuitive notion of set which we already regard as defective? That is, the conflict between (i) and (ii) might arise as a result of an underspecification in our natural language use of the term “proper subset”. So one response to this would just be to say that in the infinite case the elements of the two sets – the odds and the natural numbers – don’t really stand in the relevant proper subset relation. This isn’t entirely crazy, I think, since it’s a bit weird to say that one infinite set is a proper subset of another. It at least stretches the image we have of sets as circles containing and intersecting with other circles. ]]>

Of course, nothing I’ve said lets us compare two disjoint infinite sets (like the odds and evens). And if you’re really going to say that there are more evens greater than 0 than evens greater than 2, then that seems fine, since it’s not clear which of these sets (if either) should have the same size as the set of positive odds.

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