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In the incompleteness theorem, how do we know that "I'm not provable" may apply to a math conjecture other than the Godelian sentence itself?Neither Gödel’s Theorem nor the likes of the Liar Paradox actually demonstrates that there must be true but unprovable statements.What Gödel’s Theorem suggests is that, using a set of rules to design board games in which a tie is no allowed, such a statement as “No board game may yield a tie” would be *absolutely* unprovable, even though it’s perfectly provable within our *chosen* set of rules.The paradox-like is quite similar, what “I’m not provable” truly suggests is that, assuming this statement is semantically meaningful then we can neither prove it true nor false, whereas assuming it’s semantically absurd then we may prove it either ways on an arbitrary basis.The morale is, there is not a general conclusion that there must be true but unprovable theorem. Rather it’s that for any theorem to be true it must conform to a valid system of proving theorems, which can be any arbitrarily contrived system so long as being consistent.
Isn’t the metamathematical statement “I’m not provable” very unlike a normal conjecture or theorem? Is this not a pseudo-statement? How can it lead to a general conclusion that there must be another absolutely true but unprovable theorem?A2A. Thanks for the question, a quite interesting one, Joseph. The first thing to understand what is going on to grasp the extraordinary idea of Gödel of the arithmetization of the metalanguage of arithmetics. By means of his technique, many sentences that speak about arithmetics became also arithmetic sentences, such as "to be a variable", "to be a formula", "to be a proof of a [given] formula", "to be consistent", "to be provable", "not to be provable", and this is due in such a way that a sentence can speak also of itself, as for instance, a sentence [math]S[/math] can say "[math]S[/math] is not provable", that is, "I'm not provable". Thus this last one is not a pseudo-statement, as you have guessed. It is a legitimate arithmetic sentence. Then, by assuming certain constraints, such as that a theory is consistent, strong enough to express at least part of Peano's arithmetics, its language is recursive and so on (the right conditions were provided in another post of mine, links to: Décio Krause's answer to How does Gödel's incompleteness Theorem influence the confidence we have in the validity of scientific knowledge gathered thus far?) then there exists an arithmetic sentence, namely, [math]S[/math], which neither it nor its negation can be proven in the arithmetics (supposed satisfying the "right conditions"). So, what the first theorem says (in the form later extended by others such as Kleene), is that every theory satisfying the right conditions presents undecidable sentences such as [math]S[/math]. Notice that this theorem does not speak of truth. It speaks of demonstrability instead. The "right conditions" are essential; almost of the misunderstandings about Gödel's theorems come from the disobedience of these conditions.
How do you file an election 83(b) form/statement to the IRS for a foreign founder that has no SSN? Do we have to somehow register these founders with the IRS before they are granted restricted stock?Okay, I'm going to answer my own question here now that I know the answer.First of all, foreigners technically don't need to file an 83(b) since they are not liable for US taxes. If there is no chance that those foreign founders will ever live in the US during the vesting period, you can safely forego the 83(b)However, you should file the 83(b) anyways by putting "Applied for" in the field where the 83(b) election asks for the SSN/Taxpayer Identification Number. Technically, you don't even need to apply for an SSN/TIN until the founder actually needs it (i.e. they move to the US to work for the startup there). All you need to defend the 83(b) election in the case a founder becomes subject to US taxes is the stamped/signNowd 83(b) election back from the IRS. You can just hold on to that in case you ever need it and whenever you do happen to file for an SSN/TIN, you simply include it with your SSN/TIN filing. So, in summary, you should:1) File the 83(b) election for everyone within 30-days of the equity/options grant, always, placing "applied for" in the space where you'd normally put the SSN/TIN2) Hold onto the IRS stamped/signNowd 83(b) election form that you get back from the IRS. 3) If the founder ever does apply for a US SSN/TIN during the vesting period, include the stamped/signNowd 83(b) election among all the paperwork being submitted.Disclaimer: I'm not an attorney, but I got this advice directly from one of Silicon Valley startup lawyers that was very highly upvoted here: Who are some of the best startup lawyers in Silicon Valley?
If you work for yourself doing government contracts and American Express asks for you to show them a current pay stub, how would you provide that? Is there a form that has an earnings statement that you can fill out yourself?It seems to me you should just ask American Express if they have form you can fill out. It seems odd they would want to see an earnings statement, but if you need to show some sort of proof of income, typically in the absence of a pay stub, your most recently-filed tax return should suffice.I'd really ask them first before automatically sending them your tax returns though.