I could explain why, but you wouldn’t have enough ordinals in your brain to process this information. If you saw what I saw, you’d go insane. You’d think 0=1 and your whole life would be a walking contradiction.

No matter. What I can do is place what YOU think is an uncountable order. In fact, I can do that in only omega many steps. Oh sure, I know what you’d do. You’d process the order sequence I gave you, expand your worldview, and then realize that what you thought was omega_1 was actually countable all along. But then I can place another uncountable order again. And again. And again after that.

If we repeat this process omega many times, you’d lose your sanity. I could refer you to my psychologists, Dr. Solovay and Dr. Tennenbaum, to help you make sense of this all, but I’ll save you the trouble.

Two scoops of chocolate chip cookie dough please.

]]>I strangely wasn’t notified of your answer… Anyway, okay thank you I got it! Now in my mind occurred the switch of perspective to your point of view, and I agree not to agree.

Nice illustrative example by the way.

Would you know whether that’s the first study of topological structures that are not merely algebraic, so of compatible topological (or topological-like) enrichments of structures in general (say first – or higher – order ones*)?

Starting with, models of non-algebraic theories, such as here PA (I call them, Peanoids).

*for lack of knowledge of a more general definition of “a structure”, if there’s one somewhere…

Thank you!

]]>Cantor made the ice cream. Fodor designed the business plan, proving that there will be no catastrophic loss from one sale.

]]>That is an interesting perspective, but I don’t agree. The statement $\psi$ might not be local—if true in the universe, it might not necessarily be verifiable inside any set. But meanwhile, we are checking whether $\psi$ holds in this local set $V_\theta$. Ultimately, it is $\varphi$ that is local, because it is equivalent to the truth of a statement, $\psi$, inside a set.

Consider this: being the tallest person is not a local property, since one must make global comparisons. But being the tallest-in-your-household is local, since you just have to check if you have the being-the-tallest property in a local region, your household.

]]>I am afraid to be slightly confused by the exact meaning of this wonderful equivalence: isn’t it the property ψ which is the local one here, and not φ itself?

Thank you.

]]>The higher-dimensional analogues of the Euler characteristic are rather more complicated, and I don’t know of any simple proof by induction that follows the case for finite planar graphs. In the general case, one must pay attention to the genus of the resulting topological object. Perhaps one can prove it by induction by considering a simplicial complex. I’m just not sure, and I would welcome comments from the topologists!

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