As the crafter of the hypothetical I have to say unfortunately (or maybe fortunately depending on your predilections) your future when you come out of 'warp' looks more like this.



or maybe this



They will let you keep whatever clothes you arrive with as a consolation though.

What's the special relativity question?

This:

I don't think it quite works like that? This is basically the twin paradox by any other name, right?

Well, if you are traveling near c and make one lap around the Earth it will take less than a second Earth time. But ignoring that, you could theoretically build a spaceship that travels really fast and after 50 days on the ship 1000 years will have passed on Earth. The twin paradox is sort of an objection to that concept based on a misunderstanding of relativity.

Relativity says that physics should be the same whether you are at rest or moving at constant speed ie everybody moving at a constant velocity can take themselves to be at rest and there is no "true" rest frame that is unique or special. The twin paradox makes it seem like the Earth is actually at rest and the rocket must be truly moving because the rocket ages slower, but that's not correct.

You can see this by simply doing the reverse. Have the twins born on a rocket going near c. As they pass Earth, one twin slows down massively and spends some time on Earth then gets on another ship and catches up to his twin. In this case, the twin that was on the rocket the whole time will have aged more than the twin who was on Earth for a while. The key is the twin who experiences g forces (slowing down or speeding up) is the one who ages slower. That breaks the symmetry, not that one twin was moving and one was not.

So you could essentially time travel into the future using this method, theoretically? It works? (Well, within the framework of SL I mean).

BTW, I knew the explanation re: experiencing forces, I'm not sure that's 100% right/complete. Check out this video:



Dude is legit btw, he's a real particle physicist.

https://en.wikipedia.org/wiki/Don_Lincoln

Yes, theoretically you could see the future this way. It's not really "time travel", any more than we all time traveled from 2020 to now over the last year.

The forces is the correct/complete answer to

The forces breaks that symmetry It's not that the aging happens only during the accelerations and decelerations. I skimmed the video and I don't really disagree with what he said. It's just the next step is to ask "which twin stayed in 1 frame the whole time?". And the answer is "the twin that didn't experience any forces".

https://phys.org/news/2007-05-two-ti...dimension.html

That makes sense out of the fact that we literally feel acceleration but not velocity. That we do or why we should doesn't exactly appear obvious simply by squaring time. But along with another time dimension that empirical evidence actually warrants the possibility of and in lieu of concocting yet another physical dimension to make the equations work, another approach would be to invert our current paradigm of light moving in space.

While it is certainly intriguing, I'm skeptical of inserting theory just to make models add up.

It doesn't have to be an either/or issue. In the scheme of things with modern physics/cosmology we really don't do much relatively speaking in terms of reality research. The thing is considering the progress we have made in terms of harnessing the power of nature to do our bidding, imagine what we could do if our current paradigm is either wrong or not even wrong.

This isn't so much a puzzle, which involves a specific technique or trick not of general importance, but more an important result on the structure of logic presented in a puzzle style format.

Define a standard set of Axioms for arithmetic called A. Create a new set of Axioms called A' which consists of:

A'
1. All the Axioms of A
2. The additional Axiom that the Axioms of A are inconsistent

Is the new set of Axioms A' consistent or inconsistent? If inconsistent, show the inconsistency, if consistent explain why.

Didn't Godel already do this?

Are you basically asking someone to reformulate his incompleteness theorem?

The incompleteness theorem didn't occur to me but my intuition was that A' would be consistent but entirely useless as it couldn't be used to prove anything.

In what way does adding Axiom 2 change what followsed from A?

Also if you can't prove A is consistant from the Axioms of A then is there an inconsistancy in A'?

I am not really familiar with axiomatic systems at all but my understanding is that inconsistency is specifically the ability to prove both something and its negation, so my thinking was that if one of the axioms was that the rest of the set were inconsistent you couldn't formally prove anything. Hence consistent but useless. I certainly wouldn't be surprised if my understanding is completely wrong though.

Consistant means you can't prove both x and ~x

If it's inconsistent then you can prove both x and ~x.

but (x and ~x) is false.
and (false -> y) is always true

Hence you can prove everything from an inconsistent set of axioms rather than nothing.

It's a direct consequence of the incompleteness theorem, though you could say if you understand the theorem you can answer my question.

Right This is the key point. I told you A was consistent and a theory only proves it's consistency if and only if it is inconsistent. So A' must be consistent and even though it claims itself inconsistent, and it can't actually produce these inconsistencies in A.

It's of course totally useless. While it is consistent, it fails to have higher order conditions required for a useful theory that Godel called ω consistency.

Oh, yeah? Then prove Bertrand Russell is the Pope, smarty.

I think you actually missed the bolded out of your initial post, but I guess the question is only interesting under that assumption anyway (if A is inconsistent then A' is necessarily inconsistent).

Having read a bit more on the topic, particularly the ω-consistent theory ecriture linked, I think the distinction is in being able to syntactically prove a specific contradiction. If A is consistent then the addition of the extra axiom creates intuitive inconsistency but doesn't allow for a syntactic proof of a specific contradiction. It is therefore syntactically consistent but ω-inconsistent.

I didn't directly state it, I meant to. But I did say it was a standard set of axioms for arithmetic, which implies consistency.

Maybe we were supposed to assume that becaus it's an arithmetic but it certainly cuts down the options to ponder.

If we suspend belief as to whether the extra axiom is true or false then can't we still deduce exactly the same results from A' plus a load of uninteresting ones that include the extra axiom which we can easily ignore.

Suppose the extra axiom was that all the theorems deduced from A are blue - does that make A useless? or is the same as A plus a load of uninteresting ignorable theroems that include blueness and are intuitively false.

Sure

(x and not x) -> BR is the pope

So here you have to be careful here in terms of which theory you are talking about. Lets say A is Peano Arithmetic, so it's model is the standard integers. A' adds the additional axiom that PA is inconsistent. The model of A', however, is not the standard integers. So your statement that A' cannot prove anything is correct, in the sense that it can't prove anything meaningful about the standard integers in a non trivial way ie simply ignoring the last axiom and proving it in PA. Of course A' can prove anything and everything about the nonstandard integers that form a model of A'.

Yeah it was half rhetorical. And "If tautologies are false then tautologies are true" is true as well. One thing that jumps out is that the FT conditional can't be used to produce a sound modus tollens/ponens argument. So it's just kind of out there but harmless.

it's tough to prove MP

Proof would be of the form:
Arg -> MP
Arg
therefore MP