All is countable

From: Eliezer Yudkowsky (
Date: Mon Aug 16 2004 - 08:37:14 MDT

Christian Szegedy wrote:
>> I'm still learning, but it looks to me like physics describes
>> relations that hold between points in quantum configuration space.
>> Similarly, an axiom set, if we map it to a model, describes relations
>> that hold between elements of the model. This is what suggests to me
>> that the TOE of physics might be analogous to an axiomatic system.
>> For me the fascinating thing is that although mathematicians can get
>> along just fine using nothing but axioms, we appear to actually
>> *exist* in the model, not the axioms. I am still trying to wrap my
>> mind around the Lowenheim-Skolem theorem as it applies to this - would
>> there be any conceivable physical test that would tell us we were
>> really in a uncountable model, or would the Lowenheim-Skolem theorem
>> rule that out?
> Of cource, the first step would be to define what is "physical
> experiment" inside a model (in the sense of mathematical logic). If your
> axiom-system is suitably large (that is: it allows for simulating a
> Turing machine), then it is necessarily non-complete. Even in this case,
> you can reduce the physically sensible manifestation of the model by
> restriciting the notion of physical experiment.

Only a mathematician would reply thus! Anything I can take a bunch of
atoms, do, and actually observe, is a legitimate experiment. The
philosophical conundrum would be whether it constitutes a legitimate
physical "experiment" to ask, for example, "Does this process *ever* halt?"
and not just the experiments "Does this process halt after finite time T1,

> You may be able to reduce the notion of physical experiment so skilfully
> that the outcome of each experiment can be decided by the axioms alone.

The obvious restriction would be physical experiments performed in bounded

> Of course, in this case you cannot tell using experiments in which model
> you are.

Let's say that we use a set of axioms that can be satisfied by both the
reals and complex numbers, and the axioms uniquely determine the result of
any operation applied - analogous to laws of physics with complete
determinism and causality. Or to be more exact, the axioms uniquely
determine the relation of an object returned by an operation to any other
objects that interact with it - any experimental measurement we can perform
will have a result specified by the axioms and the initial conditions. The
experiment "square a particle, and the particle equals negative one", i.e.,
1 + square(particle) = 0, tells us that we are in the universe of complex
numbers. Ah, you say, but then the axioms do not determine the result of
the experiment:

> Otherwise the theory is not complete: it does not describe the
> physical laws completely.

You can have a set of physical laws that don't completely specify the
result of every experiment "by itself", only the result of every experiment
given the exact physical setup - the physical data of the initial
conditions. But the initial conditions are data of the model, like
pointing to a particular integer in Peano arithmetic (say S(S(S(0)))) that
wants to look at the integers around it.

You can say a good physical theory should also specify the boundary
conditions at the initiation of time (the 0), thus determining the entire
model, including the fact that I will someday perform a certain experiment.
  So the question is whether there can be a physical theory that is fully
deterministic including determining boundary conditions, where the folks
inside can perform an experiment *from within the model* and say, "Ah, we
live in an uncountable universe."

Maybe to specify the boundary condition of an uncountable model, the
physics must include what a human mathematician would call second-order
statements. The Lowenheim-Skolem theorem only applies to first-order
statements. I am told that, e.g., the least upper bound property of the
reals can only be specified with a second-order statement.

Eliezer S. Yudkowsky                
Research Fellow, Singularity Institute for Artificial Intelligence

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