I've recently learned about Gödel numbering and I have to admit; it's my favourite application of this theorem. Sorry RSA.
You don't actually need FTA to perform Gödel numbering. ASCII will do just as well.
Why isn’t the fundamental theorem of arithmetic obvious? (2011)
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Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#12"The fundamental theorem of arithmetic can be approximately interpreted 3 * 5 * 13 and 3 * 13 * 5" - this is false. The fundamental theorem of arithmetic "can be more approximately interpreted as " 195 has one prime factorization and it only includes one 3, one 5, and one 13 and no other primes. Once we find one prime factorization we want to prove that 195 does not have a prime factorization that includes another 11…
It doesn't say that. What it says is: The fundamental theorem of arithmetic states that every positive integer can be factorized in one way as a product of prime numbers. This statement has to be appropriately interpreted: we count the factorizations 3x5x13 and 13x3x5 as the same, for instance. That's not the same thing at all.
I miss quoted the original blog by replacing "has" for "can" and such. But the second sentence of what you quoted is misleading because that's not an correct interpretation of FTA.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#13Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#14I remember reading a proof to the fundamental theorem of arithmetic (every number is composed of a unique multiple of primes) in a number theory book and really enjoying it. I would disagree with Gowers and say it is obvious intuitively, but I'd still argue it is worth writing a proof for.
It really isn't obvious (I promise!), and Gowers's Answer 4 is "the real reason" for a mathematician. Alternatively viewed, it is not obvious that Euclidean domains are unique factorisation domains. (Though it is obvious that EDs are principal ideal domains, so the alternative view becomes "it is not obvious that PIDs are UFDs".)
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#15Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#16I remember reading a proof to the fundamental theorem of arithmetic (every number is composed of a unique multiple of primes) in a number theory book and really enjoying it. I would disagree with Gowers and say it is obvious intuitively, but I'd still argue it is worth writing a proof for.
The link misinterprets FTA.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#17Earlier quoted context omitted.
It doesn't say that. What it says is: The fundamental theorem of arithmetic states that every positive integer can be factorized in one way as a product of prime numbers. This statement has to be appropriately interpreted: we count the factorizations 3x5x13 and 13x3x5 as the same, for instance. That's not the same thing at all.
I miss quoted the original blog by replacing "has" for "can" and such. But the second sentence of what you quoted is misleading because that's not an correct interpretation of FTA.
> But the second sentence of what you
> quoted is misleading because that's
> not an correct interpretation of FTA.
... we count the factorizations 3x5x13
and 13x3x5 as the same, for instance.
So I still think you are mistaken, and I don't understand why you are disagreeing with what Gowers wrote. Perhaps you could give more detail about why you think he is wrong.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#18Why isn't 2+2==4 obvious? http://us.metamath.org/mpegif/mmset.html#trivia
Perhaps you should read Principia Mathematica [1], which proves on page 379 that 1+1=2.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#19"The fundamental theorem of arithmetic can be approximately interpreted 3 * 5 * 13 and 3 * 13 * 5" - this is false. The fundamental theorem of arithmetic "can be more approximately interpreted as " 195 has one prime factorization and it only includes one 3, one 5, and one 13 and no other primes. Once we find one prime factorization we want to prove that 195 does not have a prime factorization that includes another 11…
He never used the word "approximately", and I think you are reading the statement(s) incorrectly because of that. Let me be more specific.
The article says:
The fundamental theorem of arithmetic
states that every positive integer can
be factorized in one way as a product
of prime numbers.
This statement has to be appropriately
interpreted: ...
... we count the factorizations 3x5x13
and 13x3x5 as the same, for instance.
Reading your other comments, it appears you are arguing against something that you actually agree with, and I suspect you have mis-read or mis-interpreted what Gowers has written.
Re: Why isn’t the fundamental theorem of arithmetic obvious? (2011)
#20Earlier quoted context omitted.
Where did you read that?
https://www.google.com/url?sa=t&source=web&rct=j&url=http://... note that prime factorization is unique up to order. We aren't talking about order when we use the term uniqueness.
... which is exactly what Gowers is saying in the second sentence.