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lugita15@xxxxxxxxx wrote:
Aatu Koskensilta wrote:
How is the rank of a set variable in a formula in the language of set
theory determined?
That's the point. Instead of considering TST which is an obvious typed
theory, let us instead consider Zermelo set theory with the seperation
scheme restricted to formulas wih bounded quantification. The same
difficulty as in ZFC is how to assign types (ranks) to the variables.
That is the motivation for a typical ambiguity scheme; it merely relies
on the obvious intuition we have that as long as types (ranks) can be
assigned consistently for the variables in a formula, it does not
matter what particular type (rank) each variable in each formula is
assigned.
I'm not sure what you're asking, anymore. Are you asking something not
about ZFC and sentences in the language of set theory, but about some
bounded version of ZFC and sentences with only bounded quantifiers?
A correct example of typical ambiguity is that for each ordinal n, the
following is a theorem: there exists a set which contains all sets of
rank n.
Given an arbitrary ordinal alpha there is in general no sentence
expressing "there exists a set which contains all sets of rank alpha".
Yes there is. That set is called V_alpha.
V_alpha is not a sentence in the language of set theory. How do you go
from an ordinal alpha to a sentence P_alpha in the language of set
theory that expresses "there exists a set which contains all sets of
rank alpha"? Notice that there is a proper class of ordinals - most of
them not definable in the language of set theory - but only countably
many sentences in the language of set theory.
My objections do not apply to the simple theory of types in any obvious
sense. For any type n the formula "all sets of type n are empty" is false.
Then by similar reasoning, shouldn't "all sets of type 0 are empty"
also be false?
No. There is only one set of rank 0, the empty set, and it is obviously
empty. All other ranks contain non-empty sets.
--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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