Definitions of parity value

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Alan Masterman
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Definitions of parity value

Post by Alan Masterman »

(1) A number is even if it is definable as the sum of n + n.
(2) A number is odd if it is definable as the sum of n + S(n).

S() is the successor function which evaluates to the number next after the number given in the argument. For example, 1 is odd because 1 = 0 + S(0).

The above definitions are consistent with and implicit in the Dedekind-Peano-Russell axioms of arithmetic and the pre-mathematical definition of the primitive natural number line as the series 0-9. As such they cannot be countered by arguments from general or post-axiomatic mathematics.

Questions:

(1) How should a philosopher of mathematics approach the problem of defining a parity value for the number 0?

(2) How to approach the problem of defining a parity value for the aleph-Ø infinity (ie the infinity of the natural number line)?

For the purposes of this question, I assume that the aleph-1 infinity (the infinity of the real number line) could not possibly have a parity value but, if there is a contrary opinion, I'd be eager to hear it!
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Halc
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Re: Definitions of parity value

Post by Halc »

Alan Masterman wrote: September 29th, 2022, 10:12 am How should a philosopher of mathematics approach the problem of defining a parity value for the number 0?
Assuming you include it in the set of natural numbers, it's even, by the definition you gave. How is this a problem?
How to approach the problem of defining a parity value for the aleph-Ø infinity (ie the infinity of the natural number line)?
Aleph-Ø is not a natural number, so it has no parity.
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Pattern-chaser
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Re: Definitions of parity value

Post by Pattern-chaser »

Alan Masterman wrote: September 29th, 2022, 10:12 am (1) How should a philosopher of mathematics approach the problem of defining a parity value for the number 0?
Is this the "parity value" I am familiar with as a firmware designer? [See here.] Or is this something different?
Pattern-chaser

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Alan Masterman
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Re: Definitions of parity value

Post by Alan Masterman »

It's something different. I am asking a question in pure mathematical philosophy. You would need to be familiar with the DPR axioms of arithmetic to address it fully.
heracleitos
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Re: Definitions of parity value

Post by heracleitos »

Alan Masterman wrote: September 29th, 2022, 10:12 am (1) A number is even if it is definable as the sum of n + n.

(1) How should a philosopher of mathematics approach the problem of defining a parity value for the number 0?
Since 0 = 0 + 0, which is effectively the sum of n + n, with n = 0, zero is even in the natural numbers.
Alan Masterman wrote: September 29th, 2022, 10:12 am (2) How to approach the problem of defining a parity value for the aleph-Ø infinity (ie the infinity of the natural number line)?
While finite cardinals and ordinals are compatible for the purpose of arithmetic, transfinite ones are not.

According to the continuum hypothesis, aleph-0 absorbs arithmetic and can therefore not successfully participate in arithmetic.

Concerning the smallest transfinite ordinal, omega-0, since the initial segment of natural numbers is isomorphic with the initial segment starting at omega-0, they are effectively indistinguishable. Therefore, omega-0 will successfully inherit its properties from zero and be even.
Alan Masterman wrote: September 29th, 2022, 10:12 am For the purposes of this question, I assume that the aleph-1 infinity (the infinity of the real number line) could not possibly have a parity value but, if there is a contrary opinion, I'd be eager to hear it!
For aleph-1, or even aleph-k, the generalized continuum hypothesis suggests that the properties odd and even are undefinable for transfinite cardinals.

Furthermore, even for finite real numbers, the two rules mentioned are insufficient to define a workable generalization of the notion of parity. Therefore, their parity depends on completely new definitions for these properties, that need to be explicitly stated.

With the initial segment starting in the omega-1 transfinite ordinal being isomorphic with the real numbers, it will simply inherit its parity property from zero, according to the generalization that has been defined as mentioned above.
Alan Masterman
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Re: Definitions of parity value

Post by Alan Masterman »

Halc wrote: September 29th, 2022, 10:53 am
Alan Masterman wrote: September 29th, 2022, 10:12 am How should a philosopher of mathematics approach the problem of defining a parity value for the number 0?
Assuming you include it in the set of natural numbers, it's even, by the definition you gave. How is this a problem?
How to approach the problem of defining a parity value for the aleph-Ø infinity (ie the infinity of the natural number line)?
Aleph-Ø is not a natural number, so it has no parity.
It is true that 0 = 0 + 0 is allowed in post-axiomatic arithmetic. It also contradicts the rule we apply in the case of every NON-zero number; the goalposts have moved. In number theory it contradicts the Law of Identity; in set theory, we would say that the null set is indivisible. So why does arithmetic allow 0 + 0 = 0? My own hypothesis, previously expressed elsewhere, is that the sum is not really 0 but ø: a null value indicating that the operation of addition has failed. 0 + 0 = 0 appears to work only because arithmetic cannot distinguish between ø and 0. In fact, it couldn't care less HOW many 0's are equal to 0, and you will need to explain how you get a proof of evenness out of that. You also need to explain how 0 can be equal to half of itself, in spite of the Law of Identity. (There's more to this "philosophy of mathematics" than meets the eye! Arithmetic is not a logically watertight system; it allows us to get away with a lot of things which number theory frowns upon, because it doesn't contain within itself the necessary critical tools).

As to the aleph-null infinity: not so fast! It's also known as a "countable" infinity, meaning that its extreme value is, in fact, a natural number (even though we can never know what that number is). One can play with the concept in various ways. For example, the infinity of the natural numbers >=1 would seem to be an even number, because it must embrace equal quantities of even and odd numbers, and the conditon n + n is met. But then, if we add 0 to either group of numbers, the condition n + S(n) would seem to be met. This doesn't prove anything, actually, exept that we should always follow the arguments where they lead, step by step, and resist the temptation to reply by shouting received orthodoxy in a very loud voice.
Alan Masterman
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Re: Definitions of parity value

Post by Alan Masterman »

Utterly irrelevant aside to Pattern-Chaser: motto of *** Recon Squadron, Australian Regular Army: "Who stares, wins".
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Halc
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Re: Definitions of parity value

Post by Halc »

Alan Masterman wrote: October 5th, 2022, 7:27 am [0 = 0 + 0] also contradicts the rule we apply in the case of every NON-zero number;
The rule you use on every non-zero even number is:
"A number is even if it is definable as the sum of n + n." We did exactly that with the zero, so it doesn't contradict what we did with the other even numbers.
In number theory it contradicts the Law of Identity
It does? How? Does number theory assert that half any number is necessarily a different number? If not, what exactly offends the law of identity?
[aleph-Ø] is also known as a "countable" infinity, meaning that its extreme value is, in fact, a natural number (even though we can never know what that number is).
That's a crock. That's not what countable infinity means at all. You're just contradicting your own nonsense that you're seemingly making up on the fly. You come up with all sorts of contradictions in you post following that assertion.
heracleitos
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Re: Definitions of parity value

Post by heracleitos »

Halc wrote: October 5th, 2022, 6:51 pm
In number theory it contradicts the Law of Identity
It does? How? Does number theory assert that half any number is necessarily a different number? If not, what exactly offends the law of identity?
By the way, transfinite numbers may actually not always satisfy the law of identity:
Victoria Gitman wrote: In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.
While the standard model of natural numbers has a unique order, some transfinite, nonstandard models may actually have nontrivial automorphisms:
Victoria Gitman wrote: Also, while the natural numbers obviously have no nontrivial automorphisms, there are countable nonstandard models with continuum many automorphisms.
But then again, such indiscernible transfinite numbers may or may not exist:
Wikipedia on "zero sharp wrote: In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe.

The condition about the existence of a Ramsey cardinal implying that 0# exists can be weakened.

The existence of ω1-Erdős cardinals implies the existence of 0#.

Chang's conjecture implies the existence of 0#.
Alan Masterman
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Re: Definitions of parity value

Post by Alan Masterman »

There is just so much in this thread which needs replying to, but I only have one lifetime.

'According to the continuum hypothesis, aleph-0 absorbs arithmetic and can therefore not successfully participate in arithmetic"

Aleph-null is perfectly capable of participating in arithmetic, though not according to all of the same rules which apply in classical arithmetic. Consider the set of all natural numbers (A), from which we subtract the set of all natural numbers which are >=1 (B). Thus A - B = 1, while A + B = A = B. Many mathematicians prefer to avoid such contexts because their training emphasises mastery of a received wisdom rather than true critical thinking.

The main difference between infinity-arithmetic and classical arithmetic is that, whereas in classical arithmetic it doesn't matter what you are counting (1 + 1 always equals 2), in infinity-arithmetic you often cannot know the correct answer unless you first know what it is that you are counting.
Alan Masterman
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Re: Definitions of parity value

Post by Alan Masterman »

As to 0 = 0 + 0: this is possible in arithmetic because the logical rules of post-axiomatic arithmetic are rather lax in this area. According to the axioms, n = n + n is forbidden. Set theory (number theory) concurs. But (post-axiomatic) arithmetic allows it. How do we resolve this paradox? I've offered a potential solution elsewhere, but I won't bore you with a rehash here.
Alan Masterman
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Re: Definitions of parity value

Post by Alan Masterman »

By the way, the Cantorian aleph-1 infinity is also capable of participating in arithmetic, however little we may know about its full properties. His diagonal argument is entirely an arithmetical argument and is critically dependent upon the validity of the additive operator +.
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