You only need faith in two things

Every ordinal (in the sense I use the word[1]) is both well-founded and well-ordered.

If I assume what you wrote makes sense, then you're talking about a different sort of ordinal. I've found a paper[2] that talks about proof theoretic ordinals, but it doesn't talk about this in the same language you're using. Their definition of ordinal matches mine, and there is no mention of an ordinal that might not be well-ordered.

Also, I'm not sure I should care about the consistency of some model of set theory. The parts of math that interact with reality and the parts of math that interact with irreplaceable set theoretic plumbing seem very far apart.

[1] An ordinal is a transitive set well-ordered by "is an element of".

[2] www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf

Nevertheless, the lack of exposure to such attractors is quite relevant: if there was any, you'd expect some scientist to encounter it.

Yeah, it's hard to phrase this well and I don't know if there's a standard phrasing. What I was trying to get at was the idea that some computable ordering is total and well-ordered, and therefore an ordinal.

Well, supposing that a large ordinal exists is equivalent to supposing a form of Platonism about mathematics (that a colossal infinity of other objects exist). So that is quite a large statement of faith!

All maths really needs is for a large enough ordinal to be logically possible, in that it is not self-contradictory to suppose that a large ordinal exists. That's a much weaker statement of faith. Or it can be backed by an inductive argument in the way Eliezer suggests.

My guess is that deduction, along with bayesian updating, are being considered part of our rules of inference, rather than axioms.

Consider the following toy model. Suppose you are trying to predict a sequence of zeroes and ones. The stand-in for "induction works" here will be Solomonoff induction (the sequence is generated by an algorithm and you use the Solomonoff prior). The stand-in for "induction doesn't work" here will be the "binomial monkey" prior (the sequence is an i.i.d. sequence of Bernoulli random variables with p = 1/2, so it is not possible to learn anything about future values of the sequence from past observations). Suppose you initially assign some nonzero probability to Solomonoff induction working and the rest of your probability to the binomial monkey prior. If the sequence of zeroes and ones isn't completely random (in the sense of having high Kolmogorov complexity), Solomonoff induction will quickly be promoted as a hypothesis.

Not all Bayesian inference is inductive reasoning in the sense that not all priors allow induction.

If you believe in a prior, you believe in probability, right?

I was staring at this thinking "Didn't I just say that in the next-to-last paragraph?" and then I realized that to a general audience it is not transparent that adducing the consistency of ZFC by induction corresponds to inducing the well-ordering of some large ordinal by induction.

There's an infinite number of alternative hypotheses like that and you need a new one every time the previous one gets disproven; so assigning so much probability to all of them, that they went on dominating Solomonoff induction on every round even after being exposed to large quantities of sensory information, would require that the remaining probability mass assigned to the prior for Solomonoff induction be less than exp(amount of sensory information), that is, super-exponentially tiny.

But... no.

"The sun rises every day" is much simpler information and computation than "the sun rises every day until Day X". To put it in caricature, if hypothesis "the sun rises every day"is:

XXX1XXXXXXXXXXXXXXXXXXXXXXXXXX

(reading from the left)

then the hypothesis "the sun rises every day until Day X" is:

XXX0XXXXXXXXXXXXXXXXXXXXXX1XXX

And I have no idea if that's even remotely the right order of magnitude, simply because I have no idea how many possible-days or counterfactual days we need to count, nor of how exactly the math should work out.

The important part is that for every possible Day X, it is equally balanced by the "the sun rises every day" hypothesis, and AFAICT this is one of those things implied by the axioms. So because of complexity giving you base rates, most of the evidence given by sunrise accrues to "the sun rises every day", and the rest gets evenly divided over all non-falsified "Day X" (also, induction by this point should let you induce that Day X hypotheses will continue to be falsified).

Presuppositionalism is a school of Christian apologetics that believes the Christian faith is the only basis for rational thought. It presupposes that the Bible is divine revelation and attempts to expose flaws in other worldviews. It claims that apart from presuppositions, one could not make sense of any human experience, and there can be no set of neutral assumptions from which to reason with a non-Christian.

— http://en.wikipedia.org/wiki/Presuppositional_apologetics

You pretty much got it. Eliezer's predicting that response and saying, no, they're really not the same thing. (Tu quoque)

EDIT: Never mind, I thought it was a literal question.

I actually thought this was part of a sequence, because I'm missing some context along the lines of "I vaguely remember this being discussed but now I can't remember why the topic was on the table in the first place." Initially I thought it was part of the Epistemology sequence but I can't figure out where it follows from. Someone enlighten me?

This question seems to confuse mathematical induction with inductive reasoning.

I feel like this is more of a problem with your optimism than with induction. You should really have a hypothesis set that says "humans want me to be fed for some period of time" and the evidence increases your confidence in that, not just some subset of it. After that, you can have additional hypotheses about, for example, their possible motivations, that you could update on based on whatever other data you have (e.g. you're super-induction-turkey, so you figured out evolution). Or, more trivially, you might notice that sometimes your fellow turkeys disappear and don't come back (if that happens). You would then predict the future based on all of these hypotheses, not just one linear trend you detected.

If you have a method of understanding the world that works for all problems, I would love to hear it.

So induction gives the right answer 100s of times, and then gets it wrong once. Doesn't seem too bad a ratio.

I don't have faith in induction. I happen to be the kind of monster who does induction.

But only sometimes. The more toothpaste I have extracted from the tube so far, the more likely it is to be empty.

I've long claimed to not have faith in anything. I'm certainly don't have "faith" in inductive inference. I don't see why anyone would have "faith" in something which they are uncertain about. The need for lack of certainty about induction has long been understood.

Which of these seven models of faith do you have in mind when you use that word (if any)?

Unless I've missed something, it is easy to exhibit small formal systems such that the minimum proof length of a contradiction is unreasonably large. E.g. Peano Arithmetic plus the axiom "Goodstein(Goodstein(256)) does not halt" can prove a contradiction but only after some very, very large number of proof steps. Thus failure to observe a contradiction after small huge numbers of proof steps doesn't provide very strong evidence.

How?

You can do it with the axiom of choice, but beyond that I'm pretty sure you can't.