Excellent summary of infinite ordinals! Quick question, though.
I might have misunderstood your meaning, then, in my other reply to you.
It was my understanding that the set of all natural numbers had a cardinality which was equal to its ordinality. I had thought that "n" becomes "less than" ω only when it's the case that ω+1, or ω*2, or ω2, or ωω? I do remember you saying that cardinals and ordinals are different. Is it that an ordinal and a cardinal could have the same number, but that it's a different kind of number, due to the fact that ordinals and cardinals are different kinds of numbers? Thank you for your patience.
Now, let's consider the set which contains all of the Natural numbers, ω. Since any Natural number, n, is an element of ω, it is clear that n<ω. Thus, we have created a number which is greater than any of the Natural numbers.
It was my understanding that the set of all natural numbers had a cardinality which was equal to its ordinality. I had thought that "n" becomes "less than" ω only when it's the case that ω+1, or ω*2, or ω2, or ωω? I do remember you saying that cardinals and ordinals are different. Is it that an ordinal and a cardinal could have the same number, but that it's a different kind of number, due to the fact that ordinals and cardinals are different kinds of numbers? Thank you for your patience.
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