Wolfgang Schwarz

Suppose ZFC proves its own inconsistency

Suppose we find a proof, in ZFC, that ZFC is inconsistent. Does it follow that ZFC is inconsistent?

On the one hand, if we could infer from ZFC ~Con(ZFC) that ZFC is inconsistent, we could contrapositively infer the consistency of ZFC & Con(ZFC) from Con(ZFC); and since ZFC & Con(ZFC) obviously entails Con(ZFC), ZFC & Con(ZFC) would thereby entail its own consistency. Which it only can if it is inconsistent (Gödel's second incompleteness theorem). So it seems that we can only infer that ZFC is inconsistent from the observation that ZFC entails its own inconsistency if we presupposes that ZFC & Con(ZFC) is inconsistent.

Some Thoughts on Fundamental Structural Properties

Fundamental (or 'perfectly natural') properties are properties on whose distribution in a world all qualitative truths about that world supervene. That is, whenever two worlds are not perfect qualitative duplicates, they differ in the distribution of fundamental properties.

This is not the only job discription for fundamental properties. If it were, far too many classes of properties could play that role. For instance, all qualtiative truths trivially supervene on the distribution of all properties, or on the distribution of all intrinisic properties, or (for what it's worth) on the distribution of all extrinsic properties. (That's because no two things, whether duplicates or not, ever agree in all extrinsic properties.)

Structures all the way down

A structural property is a property that belongs to things in virtue of their constituents' properties and interrelations. For instance, the property being a methane molecule necessarily belongs to all and only things consisting of suitably connected carbon and hydrogen atoms.

There is two-way dependence: Necessarily, if something instantiates a structural property, then it has proper parts that instantiate certain other properties; conversely, if the proper parts of a thing instantiate those other properties then, necessarily, the thing itself instantiates the structural property.

Mistaken Intuitions

Some people intuit that

• the subject in a Gettier case has knowledge;
• Saul Kripke has his parents essentially;
• "Necessarily, P and Q" entails "Necessarily, P";
• whenever all Fs are Gs and all Gs are Fs, the set of Fs equals the set of Gs;
• the liar sentence is both true and not true;
• the conditional probability P(A|B) is the probability of the conditional "if B then A";
• it is rational to open only one box in Newcomb's problem;
• switching the door makes no difference in the Monty Hall problem;
• propositions are not classes;
• people are not swarms of little particles;
• a closed box containing a duck weighs less when the duck inside the box flies;
• spacetime is Euclidean;
• there is a God constantly interfering with our world.

They are wrong. All that is false.

Haecceities in Hilbert's Hotel

Imagine a world with nothing but infinitely many duplicate dragons, aligned in one long row. Consider the second dragon in the row. Call it "Fred".

Fred could have failed to exist. There are many worlds where he does not exist. (The actual world is probably one of them). Some of these worlds where Fred does not exist contain no dragons at all, others contain some of the other dragons in Fred's row. In particular, there are worlds where all the other dragons exist, but not Fred. The dragons are after all distinct existences, and there are no necessary connections between those.

Semantic Values and Rabbit Pictures

Robbie Williams pointed out that in my recent musings on worms and stages, I ignore the following straightforward characterizations:

Worm Theory: the semantic value of predicates like "rabbit" is a set of 4D worms.

Stage Theory: the semantic value of predicates like "rabbit" is a set of 3D stages.

He's right. I believe that these theories both cannot work, so I don't want to define stage and worm theory that way.

Tex Test

I've set up a new server, so that now this blog should support LaTeX commands. As a test, here is what I take to be the most general version of the fixed Principal Principle:

Hm. The 'x' is a bit blurry.

Parts and Counterparts II

As usual, I ended up rewriting the entire paper when I just wanted to add some footnotes to deal with timetravel and fission. Here's the new version. I still believe there is something wrong with the conclusion (which I also still find dull), but at least the characterizations of Counterpart and Fusion Theory now look better to me.

Ned Update

Turns out that I mostly use my Ned editor for making and editing websites and web-applications nowadays. So I've just fixed some bugs (like the problem with tab characters) and added some features (like switching character encodings and a zip & download option).

Semantics for Time-Travelers

I'm somewhat stuck with the parts/counterparts paper. One of the problems is to find an acceptable semantics for time travel situations.

Part of the problem is that I'm often unsure what to say about these cases. I guess if time travel were more common, we would need some new linguistic conventions. Anyway, here are some sentences that seem true to me in the following scenario: Tina decides in 2025 to meet her younger self back in 2005. So at some time t in 2005, the younger Tina is in the living room and weighs 60 kg while the older Tina is in the kitchen and weighs 70 kg. Now, these all seem true to me: