This is our monthly thread for collecting these little gems and pearls of wisdom, rationality-related quotes you've seen recently, or had stored in your quotesfile for ages, and which might be handy to link to in one of our discussions.
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Sorry, what do you mean by this? We're talking about the cardinality of the set the measure is on; this sounds like you're talking about the cardinality of its target space? (Where values of measures are somehow generalized appropriately... let's not worry about how.) It's easy to put an order on, say, Q[t] so as to make t infinitesimal but I don't see what that has to do with this. Or is that not what you meant?
So am I. But I may be confused about what cardinality even means in Nelson's internal set theory.
Let me give a simple example of the kind of thing I am thinking about. Consider the space of ordered pairs (a,n) where a is either 0 or 1 and n is a non-negative integer, i.e. an element of {1,2,...}. To each such pair with a=0, associate the measure M(0,n)= 1/2^n. To each such pair with a=1 associate the "infinitesimal measure" M(1,n)=M(0,n)/omega where omega is taken to be indefi... (read more)