Definitions of parity value
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Definitions of parity value
(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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Re: Definitions of parity value
Assuming you include it in the set of natural numbers, it's even, by the definition you gave. How is this a problem?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?
Aleph-Ø is not a natural number, so it has no parity.How to approach the problem of defining a parity value for the aleph-Ø infinity (ie the infinity of the natural number line)?
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Re: Definitions of parity value
Is this the "parity value" I am familiar with as a firmware designer? [See here.] Or is this something different?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?
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Re: Definitions of parity value
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Re: Definitions of parity value
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 (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?
While finite cardinals and ordinals are compatible for the purpose of arithmetic, transfinite ones are not.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)?
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.
For aleph-1, or even aleph-k, the generalized continuum hypothesis suggests that the properties odd and even are undefinable for transfinite cardinals.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!
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.
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Re: Definitions of parity value
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).Halc wrote: ↑September 29th, 2022, 10:53 amAssuming you include it in the set of natural numbers, it's even, by the definition you gave. How is this a problem?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?
Aleph-Ø is not a natural number, so it has no parity.How to approach the problem of defining a parity value for the aleph-Ø infinity (ie the infinity of the natural number line)?
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.
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Re: Definitions of parity value
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Re: Definitions of parity value
The rule you use on every non-zero even number is: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;
"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.
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?In number theory it contradicts the Law of Identity
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.[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).
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Re: Definitions of parity value
By the way, transfinite numbers may actually not always satisfy the law of identity:
While the standard model of natural numbers has a unique order, some transfinite, nonstandard models may actually have nontrivial automorphisms:Victoria Gitman wrote: In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.
But then again, such indiscernible transfinite numbers may or may not exist:Victoria Gitman wrote: Also, while the natural numbers obviously have no nontrivial automorphisms, there are countable nonstandard models with continuum many automorphisms.
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#.
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Re: Definitions of parity value
'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.
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Re: Definitions of parity value
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Re: Definitions of parity value
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