I also would like to know what the evidence is exactly that Scott is bad at math.

Short Online Texts Thread

What is the LessWrong-like answer to whether someone born a male but who identifies as female is indeed female?

The Lesswrong-like answer to whether a blue egg containing Palladium is indeed a blegg is "It depends on what your disguised query is".

If the disguised query is which pronoun you should use, I don't see any compelling reason not use the word that the person in question prefers. If you insist on using the pronoun associated with whatever disguised query you associate with sex/gender, this is at best an example of "defecting by accident".

Causal inference research.

2. A multitude of models

As a general rule, consistent theories have multiple models. Models have more consequences than the theories they model: for example, our model of the example system proves that there are only 2 men, even though this does not follow from the axioms. A sentence follows from the axioms only if it is satisfied in every possible model of S. ^4

Even the axiomatic theory of natural number arithmetic, which we would think is absolute, has multiple models. Mathematicians have agreed on a standard model (the so-called set of natural numbers), but it is easy to prove that other models exist:

Extend the theory of arithmetic (PA) with a new constant K, and the following (infinitely many) axioms.

 0 < K
1 < K
2 < K
...
65534 < K
65535 < K
...

Surprisingly, the resulting theory PAK is consistent. Proofs are finite: any proof of a contradiction in PAK would use only finitely many axioms, so there is a largest number n such that n < K is used in the proof. Therefore, K can be replaced in the proof by n + 1, yielding a proof of a contradiction in PA itself! Since arithmetic is consistent, there is no proof of contradiction in PAK.

We have shown that PAK is consistent relative to ZFC. Therefore, it has a model. A model of PAK is a model of arithmetic, but it is clearly not the standard model. Therefore, arithmetic has a non-standard model, which contains the standard integers, as well as non-standard integers (such as the one corresponding to our constant K). In a sense, the non-standard models contain "infinite" numbers that the model cannot distinguish from the real, finite numbers.

The existence of non-standard models is a serious issue: There are situations where the standard model has no counterexamples to a statement, but some non-standard model has. This means that the statement ought to follow from the axioms of arithmetic, but we cannot prove it because it fails in a weird, non-standard model.

For example, some non-standard models disagree with the following statement (the Ramsey theorem), which is satisified by the standard model.

For any non-zero natural numbers n, k, m we can find a natural number N such that if we color each of the n-element subsets of S = {1, 2, 3,..., N} with one of k colors, then we can find a subset Y of S with at least m elements, such that all n element subsets of Y have the same color, and the number of elements of Y is at least the smallest element of Y.

Another, more accessible example is whether you can kill the Hydra or not. You can kill the hydra in the standard model, but many non-standard models disagree. If you would number all the hydras in a non-standard model, the counterexamples would be numbered by non-standard numbers such as K in the proof above.

We need to add new axioms to the axiomatic system of arithmetic, so that it corresponds more faithfully to the standard model. However, our work is never over: as a consequence of Gödel's incompleteness theorem, new axioms can rule out some non-standard models, but never all of them.

3. Generalised models and Hamkins' paper

So far:

• Consistency relative to ZFC is a useful notion: giving a model allows us to prove that our theories are as consistent as mathematics itself.

• Arithmetic has multiple models. There is a so-called standard model of arithmetic, which is not some real-world or transcendent notion. It is merely a set that mathematicians have agreed to call the standard model. The axioms of arithmetic are unable to exactly describe the standard model: they always describe the standard model plus some other "junk" models.

Do we know that ZFC is consistent? The short answer: we don't and we can't. By Gödel's incompleteness, if ZFC is consistent then it has no models. However, by adding new axioms to ZFC (e. g. large cardinal axioms). we can create set theories that have generalised notions of models. While ZFC has no models, it does have generalised models.

Unlike arithmetic, ZFC itself has no agreed-upon standard generalised model. There is not even a standard system in which we construct generalised models. In all of the above, we have refused to choose a specific model of ZFC (i. e. we did not use the phrase "satisfied in a generalised model of ZFC" or any semantically equivalent sentences). We used the notion of provability in ZFC (which is absolute).

If we replace provability in ZFC with "satisfiability in some specific model", we are suddenly able to prove more properties about the standard model of arithmetic (similarly to how we can prove more theorems about numbers by passing to the standard model of arithmetic from the axioms of arithmetic). Unfortunately, it is well-known (and intuitively obvious) that if you and I choose different generalised models, our conclusions (about these previously undecidable properties) can disagree.

The paper of Hamkins collects some stronger results: our conclusions can disagree even if our chosen generalised models are very similar. For example

• There are two generalised models which agree upon the elements that constitute the standard model, yet disagree on the properties of these elements.

• There are two generalised models which agree upon the elements that constitute the standard model, agree upon the properties of the addition operation, yet disagree about the properties of the multiplication operations.

and so on... Unfortunately, the proofs of these rely on powerful lemmas, so I can't instantiate them to produce explicit examples.

Anyway, this should be enough to get you started.

Why exactly did Eliezer stop writing here and started writing his new articles (on intelligent characters in fiction, angel investing etc.) on Facebook or Tumblr instead?

What's LessWrong's collective mind's opinion on efficient markets hypothesis? From Facebook feed I vaguely recall Eliezer being its supporter, it also appeared in some of the Sequences. On the other hand, there is a post published here called A guide to rational investing, which states that "the EMH is now the noble lie of the economics profession".

I have a well-read layman's understanding of both the hypothesis and various arguments for and agains it and would like to know what this community's opinion is.