[FOM] Symmetry and Infinity

[Dmytro Taranovsky]

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 
capture.

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/010554.html and 
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 
additional insights.

Sincerely,
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm