FOM: second-order logic is a myth

Stephen G Simpson simpson at
Wed Mar 10 13:58:33 EST 1999

Robert Black 9 Mar 1999 17:35:58 
 > What is built in to second-order logic is not the hierarchy
 > generated by transfinite iteration of the power set operation, but
 > rather just the principle of separation.

No, this is incorrect.  A great deal of the transfinite cumulative
hierarchy is built into second-order logic.  (Here of course I am
referring to second-order logic with the `standard' or full-power-set
semantics, not with the Henkin semantics.  I assume that that's what
you are referring to also.)

For example, there is a simple translation of an arbitrary sentence S
in the language of ZF set theory into a sentence S' in the language of
pure second-order logic, such that S is true in V_Omega if and only if
S' is valid under the `standard' semantics.  Here V_Omega is the
cumulative hierarchy obtained by iterating the power set operation up
to Omega, the first uncountable ordinal.  We could also replace Omega
by much bigger ordinals, for instance the first inaccessible cardinal,
the first measurable cardinal, etc.

 > We are agreed that operating in first-order logic is the sensible
 > way of *studying* sets, but I still think that we *conceptualize*
 > sets using second-order notions - and that for example it's our
 > acceptance of second-order separation that justifies our adoption
 > of the (weaker) first-order schema.

You seem to be making a subtle distinction between `studying topic X'
and `conceptualizing topic X'.  I don't understand this.  Our basic
method of studying a topic is to conceptualize it.  By studying topic
X in the framework of first-order logic, we are forced to make all our
assumptions about X explicit.

In the case of set theory, this works out as follows.  To attempt to
study sets in the framework of second-order logic with the `standard'
semantics leads to an impasse where no progress can be made, because
all of the set-theoretic assumptions are hidden.  By contrast, when we
study sets in the first-order framework, we are forced to make all our
set-theoretic assumptions explicit, and this leads to spectacular
progress: Skolem, von Neumann, G"odel, Cohen, ....  This is the
history of set theory in a nutshell.

-- Steve

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