[sritchie]

Sorry, of course you're right on "Secondly". The right construction is ω, ω∪{ω}, ω∪{ω}∪{ω∪{ω}}...

For the first point, I went through the book long enough ago that I can't rebuild the proof here, but iirc the more rigorous idea is that you can construct a bijection between 1+ω and ω given the recipe I had above for how to represent numbers as sets, but you can't do it for ω+1, which is bijective with ω∪{ω}. The axiom of infinity declares that ω itself is a set, opening the door for transfinite numbers.

Better?

[bigdict]

Thanks, sorry for being pedantic. These sorts of constructions tend to trigger some kind of defense mechanism in me.

[sritchie]

No, of course you’re right to be! I owe myself another lap through this material and this is a good push…