I’m not sure I’d disagree with the idea that the system is crippled, but Number types that merge floats and integers are an abomination because IEEE floating point is one of the most bizarre, inconsistent, maddening number systems there is and you bet I’m gonna want to purely opt in to that nonsense.
Do the stupid numbers form a linear continuum? I don’t know. Are the stupid numbers complete? I don’t know. Are the stupid numbers closed? I don’t know. Are the stupid numbers well-ordered? I don’t know. Are the stupid numbers ill-defined? Apparently so. Are the stupid numbers “real math”? I don’t know what that means. Are the stupid numbers useful in any way whatsoever? I seriously doubt it. Why do you think I posted this in MPSIMS and not General Questions? Why do you think they are called the stupid numbers? Nonetheless, I think it would be interesting to try to derive some of the properties of the stupid numbers. Please feel free to reconstruct the axiomatic system of the stupid numbers and/or formulate definitions as necessary in order to create coherent statements about the properties of the stupid numbers.
No. Sorry, but that’s your problem. I’m just pointing out that your current system has an infinite number of holes in it. If 0.9999… does not equal one, then does 0.99995… - the partial solution to my first expansion - not equal one? I have no idea. I’d say you have one set (pun intended) of issues if it does and another set if it doesn’t. My sense is that you can’t abstract out one element and have that be different from all equivalent elements and still have a working system. My math is so old and rusty that I keep getting into trouble when trying to work out anything non-standard, so I may be completely wrong on this. If I am, then feel free to correct my doubts.
But as Thudlow Boink said: “The “Discussion” questions then might be phrased as “Is there a way of defining 0.999… such that…” all your axioms are satisfied? etc.” Again, 0.9999… = 1 is equivalent to an infinite number of other series. That’s where virtually every one of the doubters that come here fall apart, so I’m asking if your workaround is general or specific by extending TB to “Is there a way of defining all these other infinite series that normally add up to 1 as well?” Either your axiom system applies to them or not. What is your next step?
Maybe if we confine the axiom of stupidity to just 0.999… then maybe the stupid numbers consist of two disjoint continuums, one from negative infinity to 0.999… and another from 1 to positive infinity with an undefined void “between” the two. This might place some serious constraints on the operations of addition and multiplication.
Or maybe, again confining the axiom of stupidity to just 0.999…, we could define a new number (D) as equivalent to the expression 1 - 0.999… and this number lies between 0.999… and 1 if the stupid numbers are a continuum and is found nowhere else in the stupid number system. However, I don’t even know if the stupid numbers are a continuum or not. Again, this would probably place some serious constraints on the operations of addition and multiplication.
So 1 - 0.9999… is undefined? Is 2 x 0.9999… undefined? Is 0.9999… + 0.9999… undefined? If not, then what is the difference between 2 x 0.9999… and 0.9999… + 0.9999… ? If “For every x there exists a y such that x * y = 1” and x = 0.999 … ≠ 1, then what is y?
And 99/100 + 99/100[sup]2[/sup] + 99/100[sup]3[/sup] + 99/100[sup]4[/sup] + … has a partial expansion of 0.9999… at every point but is a different number than 0.9999…? What is one minus the other? How would we know which we are dealing with?
You see my confusion?
Maybe some of the axioms concerning addition and multiplication are inconsistent with the axiom of stupidity. I never said the system was consistent. Do we need to remove some axioms, other than the axiom of stupidity, to make the system consistent?
Also, maybe the operations of addition and multiplication in the stupid number system are vastly different from the operations of addition and multiplication in the real number system that we are familiar with. Remember, the stupid number system has nothing to do with the real number system.
I get the impression the OP is looking for something like hyperreal numbers or surreal numbers? Links to articles describing this have already been given above. If not, what then?
Maybe something like non-Archimedean local fields is of interest? They are a basis for another sort of alternative to classical real analysis.
The first thing I would like to determine is whether or not the axiomatic system of the stupid numbers is consistent or inconsistent. If the axiomatic system is inconsistent, then I don’t think we will get anywhere. If the axiomatic system is inconsistent, how can it be reconstructed to make it consistent? Without removing the axiom of stupidity, of course.
It’s consistent.
DKRP has stated the axiomatic system is consistent. Although several posters have stated the axiomatic system may not be well defined, no one has pointed out any contradiction making the axiomatic system inconsistent. Can an axiomatic system be proven to be consistent or only shown to be inconsistent? Does Gödel’s second incompleteness theorem apply here?
Do I need to add an axiom of order before we proceed any further? Can this be done and keep the system consistent?
Although the axiomatic system of the stupid numbers shares many axioms with the axiomatic system of the real numbers, I am reluctant to assume that it shares any properties with the real numbers. Should I make some assumptions, see where we get, then backtrack if it all falls apart?
I do admit that JRagon pointed out an obvious contradiction right away but I modified the axiom of the existence of the multiplicative inverse to fix that.
People have routinely proved that certain (limited) logical systems are consistent (and complete). E.g., Presburger Arithmetic.
Gödel’s proof only says that it cannot been done in general.
Compare to the related Turing’s proof of the Halting Problem: You can’t write a computer program that takes other computer programs and an input and have it output whether that program halts on that input.
Nonetheless we can prove many programs halt all the time. E.g., the standard “Hello World” program can, unsurprisingly, be proven to halt.
(There are few things more satisfying than to have an excuse to mention Presburger Arithmetic in a post.)
You may ask why the Axiom of Stupidity takes the form that it does. Ever since there have been Internet message boards, I have seen many people assert that 0.999… ≠ 1. I don’t recall anyone ever asserting that 1.999… ≠ 2, 99.999… ≠ 100, 0.333… ≠ 1/3, etc.
I have been reluctant to add axioms of order to the stupid numbers because for some unexplained reason I thought they might be inconsistent. However, from this point on let us include the following axioms:
a. If x and y are elements of S then one and only one of the following is true: x> y, x = y, or y > x.
b. If x, y, and z are elements of S and x > y and y > z, then x > z.
c. If x, y, and z are elements of S and x > y then x + z > y + z.
d. If x, y, and z are elements of S and x > y and z > 0, then ac > bc.
As I mentioned earlier, let D be equivalent to the expression 1 - 0.999… . Moving along, I propose the following:
Ynnad’s First Conjecture Concerning the Stupid Numbers
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The set S contains all of the elements of the real numbers and the element D, and
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The set S may be represented by the stupid number line which is identical to the real number line except for the existence of the number D.
Ynnad’s Second Conjecture Concerning the Stupid Numbers
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Ynnad’s First Conjecture Concerning the Stupid Numbers is true, and
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The number D appears in the stupid number line only once and is between 0.999… and 1.
If the second conjecture is not true then I suspect the stupid numbers might resemble the hyperreal numbers. If the second conjecture is true would that mean that ¼ + ¼ = ½ but ¾ + ¾ = 3/2 + D and 2 + 2 = 4 + 2*D?
Can anyone show that the axiomatic system of the stupid numbers is inconsistent or that either of the above conjectures are not true?
I am not trolling. Although it is pointless and stupid, I would like to learn more about the stupid numbers. I was hoping the Straight Dope folks would help me.
What OP is asking about are called non-Archimedean ordered fields, of which the Levi-Civita field is one example.
In an Archimedean field, it is easy to show that the infinite sum 0.99999999…, if it exists at all, must be 1. What about “power-series in the Levi-Civita field”? The phrase in quotation marks yields many Google hits but I’m neither qualified enough nor patient enough to wade through them.
The Levi-Civita field just scratches the surface of the wonders that await when you sacrifice the Axiom of Archimedes. For example, Conways’ Surreal Numbers are so dazzling to understand that they’re almost … surreal! I bought a book once that shows how Go problems that stump Go masters can be solved with the arithmetic of Conway’s surreal numbers!
It should surprise no one that surreal numbers describe the values of a certain class of games, because Conway invented them for precisely that purpose. Usually one learns about them in the course of studying game theory. For example, if the value of a game is positive, then player L has an advantage and will always win, and if it is negative then the other player will win.
In fact, if you make enough assumptions then the surreal numbers will be a universal ordered field, containing all the aforementioned Levi-Civita numbers, hyperreal numbers, etc.
You still have to be precise about what you mean by 0.999…, though. If you mean the usual limit 9/10 + 9/100 + …, which converges to 1 in the real numbers, it does not stop converging to 1 just because you allow for the existence of non-standard numbers.
Many posters have asked exactly what I mean by 0.999… . I think I understand quite well what it means in the real number system and yes it does equal 1 in the real numbers. I don’t know what it means in the stupid number system. That’s what I am trying to find out. Maybe one of the thousands and thousands of people that insist that 0.999… does not equal 1 could tell us what it means. Imagine the following conversation:
THEM (9.999… = 1 deniers): 0.999, does not equal 1.
ME: Yes it does.
THEM: No it doesn’t.
ME: Yes it does.
THEM: No it doesn’t.
ME: Yes it does.
THEM: No it doesn’t
ME: Yes it does.
THEM: It does not.
ME: (show them numerous proofs that 0.999… = 1)
THEM: It still doesn’t equal one though.
ME: Yes it does.
THEM: No it doesn’t.
ME: OK, fine then. It doesn’t equal 1. Now what?
If number A equals 1, but number B is not equal to one, it does not really clarify things to call them both “0.999…”.
Above, Derleth gives a hyperreal number which is infinitesimally close to, but not equal to, 1. If that is not familiar, working with it would be a good exercise.
I think we’re all saying that it is possible to have a working system in which a general class of instances is narrowly and rigorously defined from the ground up, but it is not possible to have a system in which one single fact is arbitrarily deemed to be incorrect, but not defined in any way.
What if someone said - as people actually have - that pi = 3 1/7? This cannot work for the same reason 0.999… cannot be different from 1. Pi is the limit of an infinite series. If you alter nothing else, any conclusion other than 3.1415… = pi just leaves holes everywhere.
You can’t declare that the limit of 0.9999… is not 1 without redefining what a limit is. And if you redefine what a limit is, you have to apply that to every instance in which we find a limit.
That can be done and mathematicians have done it. But they didn’t get there by insisting that Y is not Y as an axiom.