Yes. Regular DMT is N,N-DMT, Atkinson's Jaguar is 5-MeO-DMT. They have been referred to as "the power and the glory" respectively. 5-MeO-DMT is regarded as one of the most powerful and profound psychedelics, even when compared to N,N-DMT.
I personally like Windsurf a bit more than Cursor, but recently I've been far more productive using Claude Code with an IDE than I was using a VSC-derived AIDE.
My sweet spot at the moment is Claude Desktop with mcp servers for editing and aider --watch for quick fixes. Claude Code uses way, way, way too many tokens on the large project i work most on.
> Claude Code uses way, way, way too many tokens on the large project i work most on.
That's a very fair critique, and it makes the pay-as-you-go pricing model (vs. one of their subscription options) a completely unrealistic option for doing anything serious with Claude Code.
If PA proves that a number exists with some mathematical property - including being a Gödel number of a proof of something - then some number with that property must exist in every model, including the standard model. So there would have to be a standard number encoding a proof, and the proof that it encodes would have to be correct, assuming your Gödel numbering is.
The Gödel number for all of the standard statements in that collection will indeed exist.
But if it is an infinite collection, then a nonstandard model of PA will have statements in that collection that are not in the standard model, and they generally don't encode for correct proofs. (For one thing, those proofs tend to be infinitely long.)
We are talking past each other. I am responding to this:
"If PA proves that it proves a statement, PA cannot conclude from that fact that it proves that statement."
When you say "PA proves that it proves a statement," this usually means that it proves the existence of a Gödel number of the proof of the statement. If PA proves such a Gödel number exists, then via completeness one must exist in the standard model, and this number will be a standard natural number encoding a valid finite length proof.
If the above somehow doesn't apply to the argument you are making: how?
The existence of a Gödel number in a model of PA, does not imply the existence of a proof corresponding to that Gödel number. PA can only prove the existence of the Gödel number.
Now consider a function, definable from PA, named prove-gn. It is defined such that (prove-gn n) is the Gödel number for a proof from PA that G(n) terminates. This function is somewhat complicated, but it absolutely can be constructed. And from PA we can prove that the function works as advertised.
Suppose that we are in a nonstandard model of PA. For every standard natural n, (prove-gn n) will be a standard natural, and will correspond to an actual proof. So far, so good.
But any nonstandard natural n, and this model has many, will result in (prove-gn n) being a nonstandard natural. That Gödel number does not correspond to a valid proof. And therefore, in this model, PA will prove that it proves a statement, that it does not actually prove.
Therefore, from PA, we may prove that PA proves a statement, even though it does not prove that statement. In fact the statement may even be false. And absolutely none of this causes any contradiction or consistency problem, so long as the statement in question is a statement about a nonstandard natural number, and not a standard one.
Because there is no way from PA to determine if we are in the standard model, or a nonstandard model, from PA we cannot conclude that "PA proves that it proves" implies "PA proves". They really are different concepts. And we will cause ourselves no end of confusion if we confuse them.
I am not sure I follow and it is possible I may not be on the same wavelength, but here is another thought.
For any sentence S, if first-order PA (or any first-order theory) proves S, then that means that S holds true in every model of that theory, via the completeness (not incompleteness) theorem.
The statement "PA proves x" is equivalent to saying "there exists a natural number N which is a Gödel number encoding a proof of x." The letter "x", here, is a natural number that is assumed to encode some sentence we want to prove, that is, x = #S for some sentence S.
The above is a predicate with one existential quantifier: Provable(x) = there exists N such that IsProof(N, x) holds true, where IsProof says that N is the Gödel number of a valid proof ending in x.
If PA proves "Provable(x)", that means that the predicate "Provable(x)" holds in every model of PA. This means every model of PA would have some natural number N that satisfies IsProof(N, x). Of course, this number can vary from model to model.
However: the standard model, which has only standard natural numbers, is also another model of PA. So if PA proves "Provable(x)," and Provable(x) thus holds in every model, it must also hold in the standard model of PA. This means that there must be a regular, standard, finite natural number N_st that encodes a proof of x.
Since the standard model is an initial segment of every other model of PA, then every other model includes the standard model and all the standard natural numbers. Thus, if PA proves Provable(x), then the standard number N_st must exist in all models.
So we cannot have a situation where PA proves Provable(x) but all proofs of x must be nonstandard and infinite in length. This would mean no such proof exists in the standard model of PA - but then, via completeness, PA would not be able to prove "Provable(x)".
For every n, the function will return a function, that PA proves has a proof in PA.
Suppose that we're in a nonstandard model. For all standard n, there really will be a proof. For nonstandard n, there may or may not actually be a proof.
And so PA cannot prove from the fact that it proved the existence of a proof, that PA actually proves it.
That is very different though. You are now saying that for all n, PA proves that it has a proof the Goodstein sequence terminates for that n. But that is very different from the earlier claim:
"And, therefore, there must be a distinction to be made between "PA proves" and "PA proves that it proves"."
It seems that for each standard n, PA proves that the Goodstein sequence terminates for that n. If also proves that it proves that the Goodstein sequence terminates for all n, and so on. But this is very different from saying "PA proves that it proves the sentence 'for all n, the corresponding Goodstein sequence terminates,'" which is what I thought you were saying.
No, it is not different from the earlier claim. If you think that it is, then you were misreading me.
My earlier claim is that for each n, "PA proves that it proves that G(n) terminates". But in logic there is an important distinction to be made between "PA proves" and "PA proves that it proves". From the second, you cannot conclude the first. You must actually present the proof, and then verify it.
And the reason is this. It is true that, for all standard n, "PA proves that it proves" implies that "PA proves". But for nonstandard n it does not. Since PA cannot itself distinguish standard n from nonstandard n, PA cannot follow that implication to a proof.
(All of this is based on long conversations with my logic professor back in grad school. To the extent that I get it right, https://faculty-directory.dartmouth.edu/marcia-j-groszek deserves the credit. To the extent that I get it wrong, I deserve the blame.)
That ship has long since sailed. We live in a world of web apps with dynamically created content, AJAX, and so on. The main entry point for the app has a URL, and then within the app all bets are off. Same with framesets.
If it were that simple to sway markets through marketing, we would see Pepsi/Coca-Cola or McDonalds/BurgerKing swing like crazy all the time from "one well-placed ad campaign" to the next. We do not.
Thanks to well-placed ad campaigns, people have a very good idea what Pepsi, Coca-Cola, McDonald's and Burger King are. They also know what Siri is. And it would be similarly easy to establish Gemini as a household name.
Same here. I keep trying to figure out WTF agent that people are using to get these great results, because Copilot with Claude 4 and Gemini 2.5 has been a disastrous mess for me.
Somewhat similar, yes - but the vidor had a smaller FPGA, a smaller RAM chip, and the board contained a small supporting microcontroller as well as the FPGA.
My complaint is that I don't get what we are getting at when we reference "creativity," "cleverness," "ingenuity" and so on. The halting problem cannot be solved, in the general case, by any machine constructable in physical reality. That includes both computers and human beings. And computers can keep at it much longer than I can, come up with novel hypotheses to test much longer than I have the patience for, and can basically do anything I can do except try a bunch of random crap out of sheer frustration and eventually give up.
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