[FOM] Symmetry and Infinity
dmytro at mit.edu
Tue Dec 18 20:06:13 EST 2018
Infinity is beautiful and mysterious. But how do we demonstrate its
beauty to a finitist? And how do we decide which statements about
infinite sets are true? Here, I will argue that to a large extent,
infinity can be understood through symmetry. With respect to formal
systems, symmetry is more fundamental than infinity.
A predicativist objects to quantifying over infinite sets before their
totality has been constructed, with finitists also objecting to
unbounded quantification over integers. However, to make a good sense
(and axiomatization) of the quantification, we do not need a completed
totality. We just need the presence (or assertion) of certain
symmetries. The more symmetry we have, the more quantification we can
capture, and the better we can imitate quantifiers beyond those that we
The same symmetries manifest in different forms, including in finite
structures (imitating infinite ones), core models, and ordinal
notations. Also, unlike the featureless order type of the rational
numbers, what we have is symmetry in the presence of hierarchy.
Infinite is related to finite but sufficiently large, and by using
'sufficiently large' as a vague concept, extending it in certain (far
reaching) ways, and asserting symmetries, we can express claims about
infinity -- even beyond the language of second order arithmetic --
through statements about finite structures. The details are in my paper
"Finitistic Properties of High Complexity"
https://arxiv.org/abs/1707.05772 , but for example if we have a
sufficiently fast-growing n+1-tuple, x_0 << x_1 << ... << x_n, then a
Sigma-0-n statement with size <x_0 is equivalent to thereis y_1 < x_1
forall y_2 < x_2 thereis y_3 < x_3 ..., and axioms such as induction can
be expressed as properties and symmetries of the n-tuple.
As an aside, even use of numbers like 2^2^1000 can be regarded as a weak
form of infinity since we rely on symmetry to handle them as their
binary representations are too large to fit into the known aspect of the
observable universe. Surprisingly much can be done in theories with
limited exponentiation, including a reasonable treatment of real numbers
and a fixed number of higher types, and even Weak Konig's Lemma. See my
paper "Arithmetic with Limited Exponentiation"
https://arxiv.org/abs/1612.05941 for details, but here I will just note
a close parallel between:
* The ability to iterate exponentiation (roughly speaking, above
feasible numbers) n times.
* Cut elimination for Sigma-0-n formulas.
* The use of real numbers, and a finite (about n) number of higher types.
Beyond simple infinity, large cardinal axioms can perhaps be best
understood not as assertions of largeness but assertions of symmetry.
Many large cardinal axioms can be characterized through elementary
embeddings, and using fine structure (and especially ordinal notation
systems), embeddings (elementary or less so) are essentially
everywhere. Each step in a typical fine-structural model construction
is either a predicative definition (over the previously constructed
part), or a limit of the previous steps, or an addition of a
symmetry/embedding (coded by an extender).
As a reference, the constructible universe L captures Sigma-1-2 and some
Delta-1-3 quantification and can imitate the first order theory of V
(and further, allowing imitations of reflective cardinals of finite
orders). Higher core models capture more, while recursively large
ordinals (such as recursively inaccessible) capture less. However,
fine-structural models themselves are infinite objects giving us only a
partial understanding, with the real understanding coming (in the
future) through ordinal notation systems.
Our undestanding of reasonable ordinal notation systems for strong set
theories is only just emerging (see my paper "Ordinal Notation"
http://web.mit.edu/dmytro/www/other/OrdinalNotation.htm ), but the
general recipe is to use a framework that makes everything as simple as
possible, extract the necessary combinatorial ingredients of the
symmetries, and then combine them exactly right. The result, after the
canonical setting of gaps, gives a notation for every canonically
definable ordinal (in a given theory), essentially allowing one to see
how everything works and the ordinals are built.
Besides being built on embeddings, ordinal notation systems might add an
additional symmetry: Given ordinals alpha, beta, and gamma, is there a
preferred ordinal delta such that delta is to gamma what beta is to
alpha? In certain key cases, the answer is yes (delta depends on the
notation system), and its use (also referred as reflection
configurations) is possibly the final ingredient in my paper to reach
the desired strengths.
As for which statements about infinity are true, one approach is to have
as much symmetry as possible, except where we have a convincing
combinatorial statement to the contrary, which we sometimes do. The
axiom of choice is notorious for breaking symmetry, as evidenced in
Banach-Tarski paradox, or (for large cardinals) in the Kunen's
inconsistency. For all we know, we might find a stronger compelling
statement that refutes more large cardinal axioms, but as of now, the
evidence is that the ordinary large cardinal axioms are true. Now, while
ordinary large cardinals do not preclude incompleteness through forcing,
we can use other forms of symmetry to decide which forceable statements
are true. An example is my argument that GCH is true (
https://cs.nyu.edu/pipermail/fom/2006-May/010555.html ). I conjecture
that symmetry can be used to resolve all major incompleteness in set theory.
We can also use symmetry to extend set theory beyond the first order
theory of (V,in). V being the totality of sets makes it questionable to
use second order (or possibly, even unbounded first order)
quantification over V. However, by relying on convergence/symmetry as
the reflection properties of cardinals increase, we can introduce
reflective cardinals as certain analogs of the absolute infinity,
allowing the use of higher order set theory, as described in my paper
"Reflective Cardinals" https://arxiv.org/abs/1203.2270 and its precursor
"Extending the Language of Set Theory"
https://arxiv.org/abs/math/0504375 . kappa is reflective iff
(V,in,kappa) has "the right" theory with parameters in V_kappa, with an
appeal to symmetry to justify and axiomatize such use.
Symmetry is relevant regardless of whether one is:
- A set platonist: Large cardinals metaphysically exist, with symmetry
giving images of infinity in finite systems.
- A symmetry platonist: Using symmetry, the truth predicate of (V,in)
is unambiguous regardless of whether uncountable (or even just
non-rule-based infinite) sets actually externally exist.
- A formalist: Symmetry is a guiding principle for wonderful axiomatic
systems and constructs.
The above was just an overview of the connections between symmetry and
infinity (including my work), and I welcome questions, replies, and
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