BusyBeaver(6) Is Quite Large

I don’t get it. What’s wrong with the post? And https://arxiv.org/abs/1605.04343 is interesting, no?

> Every sentence ever spoken and every view ever looked at is also a number. It's not a freaky thing about "things like" busy beaver, it's a freaky thing about the concept of information.

I'd say that's a bit of a wrong or misleading statement. I think the correct version is "everything[1] can be encoded as a number". The concept of number is a very particular concept! It's pretty absurd to say "a screwdriver is a number" or "a word is a number". That is true for the peano axiomatization of numbers; but to me in particular, I believe numbers are a generalization (and formalization) of the idea or concept of quantity. There's a particular idea that refers to say 'two' apples, the quantity of apples. A word is not a quantity, it's a different concept. Even though each of them could be encoded as a number somehow!

[1]: Everything that we believe to be finite and of interest, that is. We don't know presently anything that could be used in reality (a music, picture, etc.) that can't in principle be encoded as a large enough number.

I think this is quite interesting, because this encoding is critical, and it completes the system. You essentially need a machine to turn things into numbers and numbers into things; and this is unavoidable. You can actually encode this machine itself with numbers! This number (which encodes this transcoding machine) can even be decoded by its own machine! But we cannot actually avoid the machine itself, some actual realization in the real world, because any number, in order to represent something, can only be translated by one "machine" (which can be essentially a computer, or a mind, etc.).

Instead of thinking of machines, you can also think of conventions. So you can have a convention that say the number '5' encodes the concept 'word', or maybe it simply encodes the string of letters "word" ("w"+"o"+"r"+"d"). But the convention interpretation isn't complete, because you still need someone, or something, to interpret this convention in practice and potentially turn concepts into reality, or simply manipulate those concepts in significant and useful ways.

Some more examples: (1) you can encode objects by describing a series of solid operations, essentially CAD modelling, so you have numbers that represent solids. The machine that interprets this number and is able to translate it for example into a picture, a series of instructions to be interpreted by a 3d printer, of performs operations (inferences) about the relevant solid model (for example, a structural analysis) is your "machine", i.e. your software, without which a number, or string of bits by itself doesn't mean anything (except the quantity associated with that binary number, perhaps), and again this encoding or number is essentially arbitrary, it could be very different. (2) a JPEG file for example encodes an image that is read by a software stack (jpeg decoder+picture viewer+operating system+display driver) and forwarded to your monitor to be viewed as a pixel array. Again the string of bits associated with any image could in principle represent anything else representable.

Information (Shannon information in particular) simply implies the encoding possible.

It's really interesting that a lot of the time we are performing essentially translations between different representations of a thing: a series of bits into states of pixels on a screen (a picture), [a series of bits] into a 3d printed object, into a visualization of an object on a screen, etc. (one way), or a reading of a camera sensor (photograph) into a series of bits, a conception of an object (3d modelling), a conception of a story (writing), etc. (the other way). We of course can (and must for them to be built of course) conceptualize those "machines" themselves (e.g. the software part), represent them in some way (our encoding), and then turn this representation into a realization of those machines (a software, a piece of hardware, or just a representation convention, etc.).

In other words, the mind or computer itself is always an integral part of the process, and information in a vacuum doesn't represent anything necessarily.

Finally, most of what we do is some kind of translation, inference, and construction -- everything to assist our lives. Of course some "machines" are capable of generating new concepts, those are very interesting "machines" :)

I can't pretend to be an expert, but I'll argue BB(7) is probably larger than Graham's number.

BB has to grow faster than any computable sequence. What exactly that means concretely for BB(7) is... nothing other than handwaving... but it sort of means it needs to walk up the "operator strength" ladder very quickly... it eventually needs to grow faster than any computable operator we define (including, for example, up-arrow^n, and up-arrow^f(n) for any computable f).

My gut feeling is that the growth between 47 million and 2^^2^^2^^9 is qualitatively larger than the growth between 2^^2^^2^^9 and graham's number in terms of how strong the operator we need is (with gramah's number being g_64 and g here being roughly one step "above" up_arrow^n). So probably we should have BB(7)>Graham's number.

>ex. "if you want the actual news, read Tucker, not Krugman" isn't an ad hominem towards Krugman.

But how you justify that could be one. If you are just attacking the person instead of their reporting I would call that ad hominem.

ZFC allows models of second order PA and proves that those models are all isomorphic. Within each model of ZFC there is no such thing as a nonstandard model of second order PA. One can only think it is nonstandard by looking from outside the model, no? What theorem of second order PA is ZFC unable to prove?

This is similar to how there are countable models of ZFC but those models think of themselves as uncountable. They are countable externally and not internally.