=================

=================

1 Answer

1

=================

Yes, that argument is nonsense. Whatever “constructed” means, there’s no reason to believe that having constructed a set is the same as having constructed its powerset. If the person making that argument wants to claim otherwise, they need to give precise definitions of the notions they’re using informally: what do “constructed” and “constructed in — stages” mean?

To drive this home, let me give one interpretation of these terms which is precise and even mathematically useful; and then show how your criticism is completely correct.

The cumulative hierarchy – the usual picture of the universe of sets – is defined as follows. V0=∅V_0=\emptyset, Vλ=⋃α<λVαV_\lambda=\bigcup_{\alpha<\lambda}V_\alpha for λ\lambda a limit ordinal, and Vα+1=P(Vα)V_{\alpha+1}=\mathcal{P}(V_\alpha). (I'm not defining this from the ground up - notice that I'm assuming that we already know what "ordinals" are - but precise definitions and details can be found in any set theory text.) Now where do we first get an infinite set? On day ω+1\omega+1! For every finite nn, VnV_n consists only of finite sets (by induction); so so does VωV_\omega (since every element of VωV_\omega is in some VnV_n, by definition). Vω+1V_{\omega+1}, however, contains the set VωV_\omega, which is infinite! When do we get an uncountable set? Well, on day ω+2\omega+2! Since VωV_\omega is a countable union of finite sets, it's countable; so every element of Vω+1V_{\omega+1} is countable. But Vω+1V_{\omega+1} itself is uncountable, and is an element of Vω+2V_{\omega+2}. Note that, under this definition of "construction" and "stage," every set's powerset is constructed exactly one day after the set itself. So indeed, constructing a set and constructing its powerset are not the same thing. Also, note that in a single day (e.g. going form day ω+1\omega+1 to day ω+2\omega+2) we can construct uncountably many things! Note that this contradicts an implicit claim in the argument quoted by the OP: in arguing that the binary tree can't be construct in ℵ0\aleph_0-many stages (and by the way, they really should be using ordinals, not cardinals, to index stages), they're assuming that we can only construct countably many things in one step. Well, this hinges on what exactly one means by "construct". Tl;dr, if you want to argue about infinite sets, you need to make your terms precise. 2 You should really learn to recognize this person and stop answering his questions. :-\ – Asaf Karagila 2 days ago I don't believe that I can argue with day ω\omega let alone day ω+1\omega + 1. To construct means to give a recursive or inductive definition for each element of the set. – user37237 yesterday