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*Post by bassam king karzeddin**Post by Markus Klyver**Post by bassam king karzeddin**Post by Markus Klyver*That's actually against the definition of what an integer is. But sure, you can declare the existence of multiple infinities if you like.

I remember that I wrote about this issue at Quora, where If we have to accept this infinity concept, then and without loss of generality the air thematic rules must be changed, where this is also would not solve any big puzzle

New rules (say only in positive sense since this can be easily extended to negative sense, where then you fall into infinite number of distinct infinities, where then there isn't any larger infinity, that would immediately invalidate its current existing concept

To explain it in integers first, and in any chosen integer base number system, then there are the known infinities with an endlessly repeated model such as (142857142857142857.....), and there are the unknown random digits infinities such as (2564897364....), or digits of (Pi), or (e), or sqrt(2)

Where then, every unique representation of endless sequence of digits must represent distinct unique infinity

But this is not at all helpful in finding exact solutions, but APPROXIMATION as we do know and practice

[6/7 = 6*10^n/7*10^n] =/= [(6*10^n - 1)/7*10) = 0.857142857142...], when (n) is becoming with infinite sequence of integer digits, sure

Thus (1/7 =/= 0.857142857142...), since the first is well defined but the endless representation is a symptom of a number that tries always and hopelessly to replace or equate the original fraction, but always and forever unsuccessfully, thus not any real number for sure

And since numbers are unique in digital representation, prime factorization and also Geometric reprsentations

You may imagine it as a ghost number that is very good for comparison

Another solid many time repeated example that implies infinity concept would never help in creating the real existence of a solution to unsolvable Diaophnintin equations

Consider the simple Diophantine insolvable equation (n^p = 2m^p),

where (p > 2) is a prime number, (n, m, k) are positive integers, where no solution exists in the whole set of positive integers, but what magicians mathematicians usually do plough, by finding large integers for (n, m) for the actually different Diophantine equation of this form: (n^p = 2m^p + k), where (k) is relatively negligible integer in comparison with large (n, m)

And give you always and forever a long RATIONAL solution (in decimal form), since there is no other way

Or the divide the original unsolvable equation as (n/m)^p = x^p = 2), hence the solution (x = n/m = 2^{1/p}), but the shame is so obvious even to any clever school student that is only an approximations that they MUST confess so openly and so loudly,

Also, remember that 2^{1/p), for odd prime (p) is not constructible number, but again can be calculated or CONSTRUCTED as an APPROXIMATION ONLY

The theme is that infinity can't produce any new theorems but fictions only, especially that is only understood by its own so funny definition being unreal number but greater than any number (so funny indeed)

There are many more clearer proofs than this fast one in my posts for sure

BKK

But there are no infinite integers.

Yes, this is my say in many posts, and oddly I assumed its existence to arrive at the contradictions so simply to convence you, but it seems that you get things backwards

As I said, that's against the definition of an integer. So there is no integer with infinite many many digits. All integers are finite.

But in mathematics, you do certainly accept it in all your fabricated decimal expansion for any number and don't tell me that decimal notation is a magical tool that turns those (you call infinite integers to real meaningful number), so remove that point or dot and see your (Pi) clearly, or simply don't ignore its mathematical representation as a ratio of two integers (N/10^m), where (m) is positive integer, and N is integer with

(m + 1) the sequence of digits, and if you insist on seeing it completely, then you must confess that you would require (m) to be with an infinite sequence of digits, so for N, and forget about that (Pi) in your mind

And don't tell me that mathematics can bring it in another form which is not constructible because it is impossible for sure

But you can introduce a NEW element and EXTEND the integers. Generally when we do this, then we are going to break things if we are not careful.

No need really to extend anything to avoid wisely more madness, otherwise we have to break our heads for sure

For example, there's no much sense in ∞ + ∞ = 2∞, so we usually define this new element so that ∞ + ∞ := ∞. Likewise, ∞ * ∞ := ∞, a*∞ = ∞ for any integer a, and so on. ∞ can be a number, but it is not a integer nor a real nor a complex number.

As you said, there is not any sense at all to make many laws and formulas about something really unreal,

And since infinity is not any number, nor anything else, then why do you really associate all your doomed so famous results or theorems with it anymore, maybe for the sweet so easy results or approximations only, never mind, but see that loudly moron that my theorem or results are mainly APPROXIMATIONS, and never EXACT, (The speech is mainly for readers)

*Post by Markus Klyver*The decimal expansion of 1/7 is very well-defined, but either you're just playing stupid or you don't want to learn anything new. I have REPEATABLY given you the formal definitions and SHOWN how they work. For sure, you are just playing dumb as fuck?

Still claiming that the decimal expansion of (1/7) is well defined, exactly like so many books, and after so many lessons, wonder!

(1/7) is only (1/7) and nothing else, but its decimal expansion is that ratio of two infinite integers (that you deny openly here), you may call it the ghost of (1/7), it is different as this (10^n - 1)/7*10^n), try it for large integer (n) to understand it, before you make your (n) with infinite sequence of digits, but truly (1/7 = 10^n/7*10^n),

And you would never understand the uniqueness of a number, or see the obvious fallacy trick they brainwashed you for sure

After all, how can I remove all that accumulated dust in your mind that was built over centuries and years in only a few posts

*Post by Markus Klyver*You can repeat your mantra about how the compass-and-straightedge construction is the only valid construction, but that doesn't make it true. You have yet to defend this very narrow-minded view and completely arbitrary rule.

As if you could solve any of my published puzzles here for mathematicians, which were about the same issue

And why should I teach you anymore once I know your aimless direction, but at any case, the answer you seek is well written in my posts here only

So, go and search for it if you really interested in learning

And one must not loose temper or becomes so nervous like real idiots to express his own anger or dissatisfactions, there is anther way called general critique that is not any personal, but can convey the theme for sure

BKK

Sigh. The decimal notation is one way to represent a real number. Consider a function f : N -> {1, 2, 3, 4, 5, 6, 7, 8, 9}. We then DEFINE the notation

f(-n) ... f(-2) f(-1) f(0) . f(1) f(2) f(3) f(4) ...

as the limit of the sum of f(n)/10^n as n -> ∞. For all partial sums, we have that they are bounded, for example every partial sum is bounded by f(0)+1. They are also increasing (though not monotonically as there could be a k such that f(k) = 0). And by the properties of the real numbers we have that EVERY INCREASING SEQUENCE BOUNDED FROM ABOVE HAS A LIMIT IN ℝ. So we know that this series will always have a limit, no matter what f is.

For example, we could consider f such that f(k) = 0 whenever k ≠ 0 and f(k) = 1 for k = 0. We would write this decimal expansion as

1 . 0 0 0 0 0 0 0 0 0 0 ...

And the series f(n)/10^n would indeed have the limit 1.

If you are using the Cauchy construction of real numbers, it's very easy to see that the series will converge in ℝ (simply because ℝ is defined such that all Cauchy sequences in ℝ converges in ℝ). You could also convince yourself about this using the formal limit definition, which I have laid out several times for you. We say that the series, with the partial sums S_n, converges to an element L ∈ ℝ if and only if

For every real ε > 0, there is a N such that whenever n > N, we are guaranteed to have |S_n - L| < ε.

Note that |S_n - L| will always be smaller than 10^(-n+1), so the choice N = (ln(10/x))/(ln(10)) will always work. So the series is guaranteed to converge in ℝ, as proven above.

It has nothing to do with "turning decimals into numbers" or "replacing real numbers with fictitious numbers". Decimal expansions are notation for series I defined and discussed above.

For 1/7 I could define a function f such that

* f(k) = 0, for all k < 1

* f(k) = 1, whenever k = 1 + 6n for some integer n > -1

* f(k) = 4, whenever k = 2 + 6n for some integer n > -1

* f(k) = 2, whenever k = 3 + 6n for some integer n > -1

* f(k) = 8, whenever k = 4 + 6n for some integer n > -1

* f(k) = 5, whenever k = 5 + 6n for some integer n > -1

* f(k) = 7, whenever k = 6 + 6n for some integer n > -1

Now, the sum f(n)/10^n will converge to 1/7 as n -> ∞, so the series have the value 1/7. THUS,

1/7 = 0.142857... Exactly. Per definition.

And also, we don't "make laws" for ∞ when we decide to extend the real numbers. We *define* a new element which satisfy a list of axioms, and we are going to use the symbol "∞" to represent it. We could have equally called it "weeb" or "querk", but the important part is that it has a number of properties it satisfies. One of them being, for every x ∈ R, we have x < weeb. And instead of calling it "weeb", we use the symbol "∞". But there's really not much to it, we define a new element that fall into the orderting and we define operations on this new element. We choose to represent it by "∞", because it has all the properties we would expect an infinite element to have. In this sense, ∞ is a number. But it's not a real number. It's a number in the extended reals.

But you probably don't understand any of this and will keep repeating your mantra: "the compass-and-straightedge construction is the only valid construction". This is something you have made up, and a position you refused to defend. So, you could reject any math that isn't compatible with a such requirement,t but you can't say it's WRONG because of that. EXPLAIN why the compass-and-straightedge construction is the only valid one, or accept that you don't have to have that limitation in your mathematical axiomatic framework.