[FOM] Symmetry and Infinity
Dmytro Taranovsky
dmytro at mit.edu
Sun Dec 30 15:25:58 EST 2018
(In reply to Matthias Eberl) There is a connection with your use of '<<'
(described in your reply and in the FOM thread "Indefinitely large
finite sets"), but where I go further is in the ability to interpret
infinity without relying on an external carrier set. Your use of
indefinitely extensible finite structures as an alternative to infinite
models is more closely related to my FOM posting "Finite Analogues of
Infinite Structures"
https://cs.nyu.edu/pipermail/fom/2005-November/009318.html . There, I
define finite quasimodels with many of the properties of actual models.
One can use a single finite quasimodel for sentences of bounded size, or
an infinite chain M_i to fully capture an infinite countable model.
Moreover, there I use a single uniform notion for being sufficiently
large relative to the formula size and the codes for its parameters; a
price for this is a restriction on how the elements are encoded, and
also that the predecessor of a positive integer in M_i might require
M_{i+1}.
I will now elaborate on certain points from my "Symmetry and Infinity"
posting.
Formalists: I previously wrote "I conjecture that symmetry can be used
to resolve all major incompleteness in set theory." While I am a
platonist, the qualification "all major" was used to state the
conjecture independently of one's philosophy. To permit referential
semantics, some formalists adopt a multiverse view, with a theory T
corresponding to all universes in which T holds. Statements like CH are
true in some and false in other universes. However, not all universes
(and consistent theories) are equally good, with some being the analog
of having a 20 story building without the 13th floor. While formally
valid (and worth investigating), one would rather not do ordinary
mathematics that way. It breaks the symmetry in the use of numbers.
Similarly, a statement implying GCH (or very unlikely, contradicting
GCH) may come to be seen as the ordinarily preferred way to do
mathematics. However, this preference (assuming it comes to pass) is
specifically for the role of set theory as an ordinary framework for
mathematics (which pragmatically adds unity and coherence to
mathematics), with other axioms being good for other roles.
Symmetry platonists: A symmetry platonist, even if agnostic about the
totality of real numbers, believes that degrees of closure/symmetry (for
appropriate constructs) form a directed system. To form a set meant to
represent the set of all real numbers, one can take a sufficiently
closed countable set R of real numbers. The closure is relative to the
resulting set, but at least for R, it appears sufficient to have closure
under pairing and under appropriate operators independent of R.
Then for sets of reals, one takes a sufficiently closed set P(R) of
subsets of R in a way that does not undermine the closure of R, and
similarly for P(P(R)), and so on, with sufficient symmetry so as not to
undermine previous levels of closure and also so as to have the right
theory (and get to large cardinals). Stronger degrees of closure of R
permit more levels of the construction, with some presumably permitting
the true theory of (V,in), and further. However, typical large cardinals
can easily be destroyed by forcing that does not add bounded subsets, so
getting there in V appears to require an especially precise (or to a
skeptic, an especially doubtful) composition of symmetry and maximality.
Use of infinity: Symmetry not only helps to explain infinity, but is
also the main reason why mathematicians have been interested in infinity
in the first place. While much of mathematics has a computational core,
symmetries such as translation, rotation, and scaling are most naturally
presented through real numbers, which are (or can be interpreted as)
infinite sets. One can do calculus with just finite sets, but
(ironically) absence of infinity would complicate and obscure the picture.
Sincerely,
Dmytro Taranovsky
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