the mathematical fallacy of infinity

Well it gives me a headache. Lol. I prefer social sciences.

   In fairness to you, because my name is really Arlon "Fairness" Staywell, there is another argument you haven't made yet.

   Let's look at it yet another way.  Does an infinite string of threes following a decimal point represent a value "equal" to that of another infinite string of threes following a decimal point?  An "intuitive" answer is that of course they are "equal."  How can they not be?  Are they not the same thing?

   Then if the fixed quantity 1/3 "produces" an infinite string of threes following a decimal why can't we recognize some other infinite string of threes likewise came from the fixed quantity 1/3?  Do you feel more like a winner now?  I practice this sort of thing.

   Here is the problem with that and it is where you might not guess.  It is in the first part that assumes that one infinite string is "equal" to another.  Recall that in the outset I said that algebraic manipulation of infinity "doesn't work most of the time."  It doesn't work because infinity never, never, never represents any fixed quantity.  One repeating decimal of all threes might be the "same" as another, but neither one of them is a fixed quantity <disappointment music, "wah wah wah">.  Each represents something short of 1/3 by an infinitesimally small amount.  Infinitesimally small is just as indefinite and not fixed as infinitely large.

   You might be wondering who cares and why <triumphant music>.  Elementary school teachers (in class anyway) and students prefer to ignore the complications.  Those who care have "rules" and those include that algebraic manipulation of indefinite numbers is also indefinite.  It is simply not allowed to add, subtract, multiply, divide or use any other mathematical operation on infinity to both sides of an "equation."  That certainly makes no sense.  It obliterates whatever was there.  Remember my observation that the difference between .9999...  and 1 is 1/∞ ?   We could all go home if only we could decide that 1/∞ is "equal" to zero, but we can't.  It's against the rules.  You may not divide by "infinity."

   More importantly as I already pointed out (knowing this game well), no one can use infinitely long strings whatever value imagined for them.  Calculators using decimal notation must abandon the approximation of 1/3 long before any "equality."  Prolonged calculations accumulate rounding errors.  Non terminating decimal notation is an imperfect representation of fixed quantities.

   Now play nice.

  

   I did indeed study calculus as well as English.  What you are attempting is an arbitrary English definition.  If I show a picture of a fish I mean three, if a picture of a monkey I mean four, if an elephant I mean five, if an infinite string of 9s following a decimal point I mean 1.  You are arbitrarily defining (often allowed) an infinite string of nines as one.  That would be fine if you understood it's arbitrary.  Apparently you don't understand.  There is nothing inherent in indefinite values that allows any use of an equal sign.

   It isn't "legal" to say 1/∞ = 0.  Notice what would happen if it were.

   If 1/x = y then 1/y = x (multiplying by x, dividing by y)

   If 1/∞ = 0 then 1/0 = ∞  Notice however that no software is going to allow 1/0 except maybe some weird version of Javascript.  No, that Javascript is not "university pure mathematics."

   Sometimes that happens even when there really isn't any more clear way to put things.

   Indefinite values are just as indefinite whether it's calculus, programming or whatever it is you're doing.

   I have a calculator that performs operations on fractions without using floating point math whenever possible.  For example to add 1/3 and 1/2 you enter the integer 1 as the numerator and the integer 3 as the denominator of 1/3.  Then you enter +.  Then the integer 1 as the numerator and the integer 2 as the denominator of 1/2.  Only integers are used in the calculation, not decimal approximations.  Then you press "exe" ("execute" not "="). and the result is 5/6 ("=" is justified here though), the fraction, not .83333 .... the non terminating decimal.  There is a button to get a decimal approximation of 5/6 if you happen to want to be like you.

   Another problem with 1/∞ "=" 0, which you obviously need to win your argument, not only does it lead to 1/0 "=" ∞, 2/0 would also "=" infinity.  100/0 would be infinity too.  Anything/0 would be infinity.  In other words fixed quantity math is abandoned.  That's why division by zero is such a terrible idea.

   I am surprised.  I would have thought even hoi polloi like you would know the difference between integer and floating point math.