In response to: Math prof: Be careful what we do with infinity. Weird things can happen: “Some weird things are like 1 = 0, not just weird, but undesirable. So we try to build our mathematical ideas to avoid those. But other weird things don’t contradict logic, they just contradict normal life,”

a reader writes,

Indeed, when you introduce ‘infinity’ into your equation, one of the (far worse than “weird” or “undesirable”) results you get is that 1=0 , which is a logical contradiction/impossibility (which is to say, utterly impossible).

Or, to put this another way, you can “prove” anything with a false premise.

And, this “result” that 1=0 is one more way we can know that the age of the universe is not infinite.

*See also:* Durston and Craig on an infinite temporal past . . .

and

Cosmologist: In an infinite multiverse, physics loses its ability to make predictions. And that’s okay.

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Hmm. I read the Eugenia Cheng post, and I believe her point was that naively treating “infinity” as a real number (for example) can lead to contradictions, hence you shouldn’t naively treat “infinity” as a real number.

Any chance this reader would be willing to elaborate on this derivation of 1 = 0?

DaveS:

Not the reader, but here’s an easy one:

Infinity + 1 = Infinity ->

Infinity + 1 – Infinity = Infinity – Infinity ->

1 = 0.

I’m going to go out on a limb here and say that infinity may be best thought of as a property. Saying something is “infinite” in general is often declaring that it’s some set whose cardinality is some transfinite number. So, saying “infinity” really isn’t enough, as you’re referencing a whole class of objects.

Beyond that, playing with any particular infinity requires you go back to how that particular infinity is defined/generated. I guess in a fashion, transfinite numbers serve as a shorthand for definitions/generators. So, if you need infinity to do something, you create a definition for infinity that allows that something to emerge.

The fun part is that “infinity” may be a concept beyond nature as it can be known, as the Big Bang might put a limit on time, globally, and quantum physics allows one to hypothesize minimum units of time and distance locally. Thus, it may very well be beyond naturalism; and, as it appears to be our only way of approaching existence as a whole, might just put that beyond naturalism. Forever. Maybe.

LocalMinimum,

I agree with much of what you say, especially concerning definitions. There are different notions of infinity, and we have to be clear about what we are working with.

This applies to the derivation you show at the beginning of the post, so I would first ask what number system you are working in. We have to know the rules of arithmetic in that system to judge whether it’s correct.

If we’re working in the extended real number system, for example, then the difference infinity – infinity is undefined, so we have a problem in the second line.

If we’re working in the class of cardinal numbers, say where “infinity” stands for aleph_null, again for example, then aleph_null – aleph_null isn’t defined uniquely. Therefore aleph_null doesn’t have a unique additive inverse, hence we can’t apply cancellation in the original equation to show that 1 = 0.

In ordinal numbers, this strategy won’t work because omega + 1 is not equal to omega (and likewise for other ordinals).

More generally, I don’t think these derivations shown in the OP or in the book by Eugenia Cheng actually can be used to show that the existence of some sort of infinity in a number system leads to a contradiction.

Okay. let’s start with this definition: infinity = infinity.

It follows: infinity – infinity = 0

Now let’s return to what LocalMinimum pointed out:

Correction to my #3, the aleph-null part. Strike the part about additive inverses, I’ll just say you can’t derive a = b from aleph-null + a = aleph-null + b.

Origenes,

infinity = infinity is not a very good definition. What number system are you working in?

Why is it “not a very good definition”? Is A = A also “not a very good definition”?

Origenes,

Well, I need to know what number system we are discussing here, so that I can decide whether your calculation is correct. The circular definition of “infinity” that you have provided does not give me that information.

I don’t get it. What number system renders A = A invalid?

Origenes,

It’s not the A = A part, but rather A – A = 0. This equation does not always hold in all number systems, for instance in the case where A – A is undefined.

If A – A is undefined, then A = A does not seem to hold.

How so?

In any case, the extended real number system is a counterexample. ∞ − ∞ is undefined in that system despite the fact that ∞ = ∞.

O, Infinity in bare form as such is more a concept than a number. In summations and integrals in effect increase without upper real-value or natural number bound. That brings in problems with subtraction etc. In sets and the like as counting numbers go to the ellipsis of endless continuation [which I contend is loaded with significance] we recognise a new quantity the order type of the counting numbers, w. We then proceed to the full surreal numbers great and small (this brings back infinitesimals) where w, w+1 etc have scale metric — cardinality — aleph-null. In effect, first type of transfinite. c, the continuum metric is another type of transfinite. Power sets in succession of aleph null can be brought to bear and there is a debate about assigning c to one in this series. For over a year I have had several exchanges with DS that pivot on not being able to traverse an endless span in finite stage countable and cumulative steps. In this mix is the point that the temporal succession process from one stage to the next which leads to the past being a pushdown stack is like that and does not have power to ground claims of an infinite past for the physical world. KF

DaveS:

I wasn’t making an assertion, I was elaborating on the statement by News that you quoted, showing how you can get 1 = 0 by treating infinity as a number. It’s clearly a contradiction, as News stated, arising from poorly defined terms, as you stated. I really wouldn’t know of anything to do with it after that.

LocalMinimum,

Thanks for the further explanation. I agree completely.

to Origenes: A – A = 0 is true only if A is a finite number. Infinity is not a number to which numeric arithmetic can be applied.

A = A applies to anything: it just says everything is identical to itself. But you can’t treat it as an algebraic equation unless A is something upon which the operation of subtraction applies, and infinity is not such a thing.

Infinity, even the “normal” kind we consider when we work with the counting numbers, has many paradoxes if you think of it as having the same behavior as finite numbers. That just means that infinity as a concept has to be handled differently than finite numbers.

For instance, the set of all counting numbers is infinite: that is a property of the set, not a “final” number in the set. The set of all even numbers is infinite. If you take away all the evens from the counting numbers you get the odds, which is also an infinite set.

Thinking of infinity as a number is just the wrong things to do, as it isn’t a number in the same sense that 5 (or 17 trillion, or 10^6000) is.

A couple of other thoughts, about multiple meanings in different contexts for the = sign.

When we write A = A as statement of logical identity, the = sign means something different than when we write, for example, x = x, where x is a number. In the latter case, the = sign means that the cardinal value of x, as a set of elements, can be put in a one-to-one correspondence with itself. The same sign is being used, but it has a different meaning in logic than it does in numerical or algebraic mathematics. Applying subtraction to the first case is inappropriate, as subtraction is not a defined operation in symbolic logic.

A real world example, which I used to discuss with my geometry classes while teaching about line segments, and especially about congruency. We make a distinction between line segment AB (written with a bar over the AB, which I can’t type here, so I’ll write as bar AB) and the length of the segment, which is a number, and is written just AB.

Some books then use a different symbol, an equal sign with a ~ over it, which I’ll type as ~=, to distinguish stating that a segment bar AB is congruent with itself to distinguish the two kinds of statements: one about congruence and one about numerical size.

For example, if you have two triangles with a common side AB that you wished to prove congruent, you would write bar AB ~= Bar AB to state that ANB would be logically identical with itself, and then go on to show that triangle ABC ~= triangle ABD. In these statements, a simple = sign about numerical identity would have no meaning.

The statement A = A has a logical statement is analogous to the ~= sign for congruence. It is a different kind of relationship than the numerical = sign.

May I recommend “Everything and More: A Compact History of Infinity” by david foster wallace.

Wow, thanks for the reference. I’ve read bits and pieces of DFW’s work but hadn’t heard of this book.

One of the courses that I took in college was “Non-Euclidean Geometry,” which involves the application of mathematical logic to strictly defined entities that exist only in mathematics. Some of the logic results in counter-intuitive conclusions.

Similarly, the definition of “infinity” might also involve entities that exist in only in mathematics, and not instantiate in physical reality.

In contrast, L’Hospital’s Rule involves ratios of the

ratesof numeric change toward extremely large numbers (infinity) or zero, while comparisons of infinities does not consider rates (see Georg Cantor, comparison of the set of counting numbers with the set of integers).However, the mathematics of

imaginary numberscan beappliedto electrical engineering (and other) problems.Perhaps the key point here involves the pragmatism of applying a mathematical system without conflating it with physical reality.

-Q