# Induction vs Rubbish

From: Russell Standish <r.standish.domain.name.hidden>
Date: Wed, 25 May 2005 10:01:04 +1000

On Tue, May 24, 2005 at 10:10:19PM +0100, Patrick Leahy wrote:
>
> This is very reminiscent of Lewis' argument. Have you read his book? IIRC
> he claims that you can't actually put a measure (he probably said: you
> can't define probabilities) on a countably infinite set, precisely because
> of Cantor's pairing arguments. Which seems plausible to me.

It makes a very big difference whether he said probability or
measure. One can easily attach a measure to a countable set. Give each
element the same value (eg 1). That is a positive measure. However, it is not a
probability, as it cannot be normalised.

One can also sample from a measure without mean - however the rules
for computing expected outcomes differs somewhat from just taking the
mean as the expectation.

For example with a uniform measure, the expected outcome is any point
in the set. Assume some property is distributed over those points -
for example the property is identical (the delta distribution). The
the expected value of that property is the constant value. and so on.

>
> Lewis also distinguishes between inductive failure and rubbish universes
> as two different objections to his model. I notice that in your articles
> both you and Russell Standish more or less run these together.
>

I'm interested in this. Could you elaborate please? I haven't had the

If what you mean by by the first is why rubbish universes are not
selected for, it is because properties of the selected universe follow
a distribution with well defined probability, the universal prior like
measure. This is dealt in section 2 of my paper.

If you mean by failure of induction, why an observer (under TIME)
continues to experience non-rubbish, then that is the white rabbit
problem I deal with in section 3. It comes down to a "robustness"
property of an observer, which is hypothesised for evolutionary
reasons (it is not, evolutionarily speaking, a good idea to be
confused by hunters wearing camouflage!)

In that case, how am I conflating the two issues? If I'm barking up
the wrong tree, I'd like to know.

Cheers

```--
*PS: A number of people ask me about the attachment to my email, which
is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.
----------------------------------------------------------------------------
A/Prof Russell Standish                  Phone 8308 3119 (mobile)
Mathematics                         	       0425 253119 (")
UNSW SYDNEY 2052         	         R.Standish.domain.name.hidden
Australia                                http://parallel.hpc.unsw.edu.au/rks
International prefix  +612, Interstate prefix 02
----------------------------------------------------------------------------

```

• application/pgp-signature attachment: stored
Received on Tue May 24 2005 - 20:28:52 PDT

This archive was generated by hypermail 2.3.0 : Fri Feb 16 2018 - 13:20:10 PST