Re: "Set Theory: Constructive and Intuitionistic ZF" on SEP

LudovicoVan <julio@xxxxxxxxxxxxx> writes:

Some of you might be interested in this article just published on the
Stanford Encyclopedia of Philosophy:

"Set Theory: Constructive and Intuitionistic ZF"
http://plato.stanford.edu/entries/set-theory-constructive/

As a side note, related to some well known discussions usually going
on in this group, here is a quote that I find "interesting" -- the
very terminology is indeed enough to conclude that this author must be
an incompetent and a liar, surely to be banned from any teaching:

Your conclusion is of course unwarranted.
The position below is basically the one mentioned by myself and Aatu
recently.

You do know that WM gets called such names not for his philosophical
stance, such as it can be made out, but for completely different reasons,
don't you?

[quote]

1.3.1 The universe of sets

The shift from classical to intuitionistic logic, as well as the
requirement of predicativity, reflects a conflict between the
classical and the constructive view of the universe of sets. This also
relates to the time-honoured distinction between actual and potential
infinity. According to one view often associated to classical set
theory, our mathematical activity can be seen as a gradual disclosure
of properties of the universe of sets, whose existence is independent
of us. This tenet is bound up with the assumed validity of classical
logic on that universe. Brouwer abandoned classical logic and embarked
on an ambitious programme to renovate the whole mathematical
landscape. He denounced that classical logic had wrongly been
extrapolated from the mathematics of finite sets, had been made
independent from mathematics, and illicitly applied to infinite
totalities.

In a constructive context, where the rejection of classical logic
meets the requirement of predicativity, the universe is an open
concept, a universe "in fieri". This coheres with the constructive
rejection of actual infinity (Dummett 2000, Fletcher 2007).
Intuitionism stressed the dependency of mathematical objects on the
thinking subject. Following this line of thought, predicativity
appears as a natural and fundamental component of the constructive
view. If we construct mathematical objects, then resorting to
impredicative definitions would produce an undesirable form of
circularity. We can thus view the universe of constructive sets as
built up in stages by our own mathematical activity and thus open-
ended.

In set theories based on intuitionistic logic, predicativity is
usually expressed by restricting the axioms of separation and
powerset, as these appear to be the main sources of impredicativity
(when the infinity axiom is assumed). [...]

[/quote]

Enjoy,

-LV

--
Alan Smaill
.

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