I mean it's logically impossible to formally and specifically define the natural numbers without introducing a logical inconsistency. The best you can do is define a set that has all the properties of natural numbers but will also define things that aren't natural numbers as well.
As an analogy you could imagine trying to define the set of all animals with a bunch of rules... "1. Animals have DNA, 2. Animals ingest organic matter. 3. Animals have a nervous system. 4. ... etc..."
And this is true of all animals, but it will also be true of things that aren't animals as well, like slime molds which are not quite animals but very similar to them.
Okay so you keep adding more rules to narrow down your definition and stamp out slime molds, but you find some other thing satisfy that definition...
Now for animals maybe you can eventually have some very complex rule set that defines animals exactly and rules out all non-animals, but the principle is that this is not possible for natural numbers.
We can have rules like "0" is a natural number. For every natural number N there is a successor to it N + 1. If N + 1 = M + 1 then N = M. There is no natural number Q such that Q + 1 = 0.
Okay this is a good starting point... but just like with animals there are numbers that satisfy all of these rules but aren't natural numbers. You can keep adding more and more rules to try to stamp these numbers out, but no matter how hard, even if you add infinitely many rules, there will always be infinitely many numbers that satisfy your rules but aren't natural numbers.
In particular what you really want to say is that a natural number is finite, but no matter how hard you try there is no formal way to actually capture the concept of what it means to be finite in general so you end up with these mutant numbers that satisfy all of your rules but have infinitely many digits, and these are called non-standard natural numbers.
The reason non-standard natural numbers are a problem is because you might have a statement like "Every even integer greater than 2 can be written as the sum of two primes." and this statement might be true of the actual natural numbers but there might exist some freak mutant non-standard natural number for which it's not true. Unless your rules are able to stamp out these mutant non-standard natural numbers, then it is not possible to prove this statement, the statement becomes undecidable with respect to your rules. The only statements you can prove with respect to your rules are statements that are true of the real natural numbers as well as true of all the mutant natural numbers that your rules have not been able to stamp out.
So it's in this sense that I mean that it's not possible to specifically define the natural numbers. Any definition you come up with will also apply to mutant numbers, and these mutant numbers can get in the way of you proving things that are in principle true about the actual natural numbers.
It seems you know what you are on about! Thank you for a cracking comment.
I've always had this feeling that the foundations (integers etc) are a bit dodgy in formal Maths but just as with say Civil Engineering, your world hasn't fallen apart for at least some days and it works. Famously, in Physics involving quantum: "Shut up and calculate".
Thankfully, in the real world I just have to make web pages, file shares and glittery unicorns available to the computers belonging to paying customers. Securely ...
The foundational aspect equivalent of integers in IT might be DNS. Fuck around with either and you come unstuck rather quickly without realising exactly why until you get suitably rigorous ...
I'm also a networking bod (with some jolly expensive test gear) but that might be compared to pencils and paper for Maths 8)
Maxatar|10 months ago
As an analogy you could imagine trying to define the set of all animals with a bunch of rules... "1. Animals have DNA, 2. Animals ingest organic matter. 3. Animals have a nervous system. 4. ... etc..."
And this is true of all animals, but it will also be true of things that aren't animals as well, like slime molds which are not quite animals but very similar to them.
Okay so you keep adding more rules to narrow down your definition and stamp out slime molds, but you find some other thing satisfy that definition...
Now for animals maybe you can eventually have some very complex rule set that defines animals exactly and rules out all non-animals, but the principle is that this is not possible for natural numbers.
We can have rules like "0" is a natural number. For every natural number N there is a successor to it N + 1. If N + 1 = M + 1 then N = M. There is no natural number Q such that Q + 1 = 0.
Okay this is a good starting point... but just like with animals there are numbers that satisfy all of these rules but aren't natural numbers. You can keep adding more and more rules to try to stamp these numbers out, but no matter how hard, even if you add infinitely many rules, there will always be infinitely many numbers that satisfy your rules but aren't natural numbers.
In particular what you really want to say is that a natural number is finite, but no matter how hard you try there is no formal way to actually capture the concept of what it means to be finite in general so you end up with these mutant numbers that satisfy all of your rules but have infinitely many digits, and these are called non-standard natural numbers.
The reason non-standard natural numbers are a problem is because you might have a statement like "Every even integer greater than 2 can be written as the sum of two primes." and this statement might be true of the actual natural numbers but there might exist some freak mutant non-standard natural number for which it's not true. Unless your rules are able to stamp out these mutant non-standard natural numbers, then it is not possible to prove this statement, the statement becomes undecidable with respect to your rules. The only statements you can prove with respect to your rules are statements that are true of the real natural numbers as well as true of all the mutant natural numbers that your rules have not been able to stamp out.
So it's in this sense that I mean that it's not possible to specifically define the natural numbers. Any definition you come up with will also apply to mutant numbers, and these mutant numbers can get in the way of you proving things that are in principle true about the actual natural numbers.
gerdesj|10 months ago
I've always had this feeling that the foundations (integers etc) are a bit dodgy in formal Maths but just as with say Civil Engineering, your world hasn't fallen apart for at least some days and it works. Famously, in Physics involving quantum: "Shut up and calculate".
Thankfully, in the real world I just have to make web pages, file shares and glittery unicorns available to the computers belonging to paying customers. Securely ...
The foundational aspect equivalent of integers in IT might be DNS. Fuck around with either and you come unstuck rather quickly without realising exactly why until you get suitably rigorous ...
I'm also a networking bod (with some jolly expensive test gear) but that might be compared to pencils and paper for Maths 8)
unknown|10 months ago
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