@Kevin suppose that we followed the backward trail and we keep going on till we reach P(base). now if I proved P(base) earlier on I'd know that the trail made me land on a value that I proved earlier and hence C (the set that refuts P(n) is empty)
but suppose that I did not prove P(base). Then if I follow the trail back I reach P(base) but then I dont know, maybe P(base) refutes the claim too
@AbdulRahman If you can show that ~P(m) -> ~P(m-a) for generic m and a then no such base case exists. It is an infinite regress. Because the proof for m and a goes through for m-a and b, etc etc.
@AbdulRahman Then there must be a problem with your proof that ~P(m) -> ~P(m-a) for all natural number m and a<m, otherwise ~P(1) -> ~P(1-a) for some a
@Peter. suppose that our supposed minimum applies that ~P(m)->~P(m-a) so we contradict that m is the minium. so essentialy the template wants to have us thinking that no such minimum can ever exist. because we proved it for a generic m
@AbdulRahman Suppose we carry out your procedure and we enumerate all the natural numbers for which P(n) is false. We have as a premise "m is the smallest natural number for which P(n) is false"
In your case, you're assuming a minimum exists, and finding $m'$ in $C$ smaller than the minimum. This contradicts the very definition of $m$. Thus $C$ cannot be nonempty.
@AbdulRahman We get all the way down to P(1). If P(1)is true, then there is STILL another natural number, m-a which is less than m and for which P(m-a) is false so that is a contradiction. If P(1) is false, then we have a number smaller than m such that P(1) is false. Also a contradiction.
@AbdulRahman So we see that the conclusion that C is empty follow regardless of whether or not P(1) is true or false
@AbdulRahman That is, it cannot be that case that both 1 and NOT 1 are the minimum element for which P(n) is false
@Peter than how about the following template to prove P(n) 1)Assume C exists which refutes P(n) 2)then there has to be a minimum m in C 3)But then P(m-1)->P(m)
@Peter in 3 what I am saying is that P(m-1) is true since m is the minimum value in the refutation set. and then we go onto prove that P(m-1)->P(m) hence C is empty and P(n) is true
@Peter/@Kevin So what about a statement in which P(base) is false. But P(m)->P(m+1) can be proved. hence there is no starting point to create the domino effect. the above proof technique will let that proof pass through
you can refer to "All horses are the same color problem" for example where P(m)->P(m+1) for m > 2
@AbdulRahman I am saying that if you assume that m is the minimum and then find that P(base) is also false that is a contradiction. So of course P(m-1) -> P(m) follows
@AbdulRahman This has no effect on the template. If P(base) is true, the template works. If P(base) is false, it still works.
@AbdulRahman Do you agree or disagree with my last statement?
@Kevin how does it work if P(base) is false. because if P(base) is false then the P(n) is false since we have a counter example but since we have been able to show P(m-1)->P(m) the template says P(n) is true. Conflicting results
@AbdulRahman I think I understand your confusion a bit better now. It has to do with keeping track of assumptions and 'discharging' them properly. If you need any further help on this let em know and we can discuss it privately.
@AlecTeal I mean it was a very basic question, and when the OP realized he posted it on a research level site, he just defaced it into some "I'm sorry didn't realize... blahblah". This is the new better looking question.
You think in the 1000s of years elves had been exploring Middle Earth that they missed the always-cloudy area with the big orange glowey thing that spews out smoke and Lava?
It's from the greek barbaros, but the Romans were great helenophiles (Love of greek culture, hence a word derived from greek) so when they imported all the Greek word, Latin got a slightly modified form. Barbari for example is the objective plural.
Greek added a lot to Latin, because Latin only really classified what one could have, for example a sailor or nauta (objective, you would assign a nauta to a boat, but a nautam is is the subjective you would say hi to a nautam).
Greek added philosophy really, like "ethos" which means "character" - something Latin had no concept of.
@KevinDriscoll if you want to know how to tell, if you spill something you can call it a spillage because you can change the quantity.
With enough pushing you can make two puddles into one bigger puddle!
With signs you can't (melting down and such is across the ambiguous line :P) hence signs.
@KevinDriscoll my biggest hate though is when - particularly Americans - I must say, no offense, add prefixes to words that already have prefixes. For example "disencourage"
Or adding wrong prefixes, "disinstall" -> "uninstall".
IIRC natio, which is Latin for "an object that was born" - or more formally "that which has /been/ born" lead to nation. @KevinDriscoll
A nation describes people, nationalism is a shared feeling over a nation, given by how many American speeches say "Americans" not "people" - nationalism is strong.
But again, go down south and it becomes hard to tell between racism and fascism, the horseshoe bends round after all, but going "American made goods FTW!" is good, a bit of nationalism, going "KKK FTW" is fascism.
@KevinDriscoll another interesting thing about the US is how the state is perceived within it. For example a guy called Stanley Milgrim did some famous experiments involving obedience.
@KevinDriscoll volunteers were made to believe they were giving an ever increasing electrical shock to another volunteer (was actually pre-recorded), yeah that.
But it was particularly strong in America, this is the classic example used at A-level for cultural bias/consistency in psychological measures.
yes there were even an experiment done where a group of people acted like prisoners and another group of people were prison guards. They had to end the experiment early because the effects were out of control.
@AlecTeal I think its interesting because there are some areas where Americans seem to want to keep government out of their lives and other areas where they want the government to force other people's obedience. I have yet to see a consistent view on this
Seriously, it's like a super power. I feel like a Sith Lord partrolling Degobah (before it all went to shit)
I dunno, if I had to say something it seems you (as in Americans) feel inferior, your grammar changes, you try to use longer words, but get all muddled and say "that would be implying" instead of "this implies".
You see @KevinDriscoll, Europe in general is a lot older than America, and after the Romans left things kind of went backwards in every sense (this is why Latin died, the common folk abused words and such to rebel against the order), transport also declined, so for many hundreds of years there was very limited migration within the country.
Nations were defined more geographically really, not by the state though, Scotland's accent starts when it gets very hilly, the north was where the coal mines and steel mines flourished, same with the welsh, the south meanwhile was more agriculture, artisan/craftsmanship and trade.
Where as America by definition of mass-migration was able to support internal migration, that is why regional differences are not as strong (remember California alone is 20 times the size of the British Isles)
So yeah @KevinDriscoll, China was always ruled by someone, as a result there was a moderate amount of internal movement, based on work, military need and such.
@KevinDriscoll if I had to make a prediction I would say that knowledge of the difference between nouns and something able to do a verb will fall further.
I was trying to explain to an American on Skype the difference between a noun and a verb, and something undergoing a verb, it was interesting because it taught me first hand the importance of language and Empiricism (that every idea we have is derived from experience, but with the addition of Kant's conceptual scheme).
@AlecTeal I'm not sure I understand precisely what you mean. Do you think that all nouns that don't act as active subjects in a sentence are being neglected or just some specific category?
He could not grasp the idea, in the same way some tribes have 50 words for what we would all just call "blue" and only one green, they can tell all the blues apart, we cannot, but they can look at green an a greeny-orange and not tell them apart.
@KevinDriscoll what's the difference between "duplicate" and "replicate"
There was a (wrong) thought that the "du" in duplicate was derived from words like "double", so if you replicated something once, it was called (wrongly again) duplication, because you made but one copy @KevinDriscoll
