I often see the belief that infinity is a quantity, when I think it's a quality. This post introduces two qualitative notions on a field where people are likely to have a good quantitative understanding in order to show how the qualitative understanding isn't that satifying.

Assume natural numbers and then we will extend them with two new numbers. They will be suggestively named "inpositivity" and "positivity".

What these numbers are is characterised by rules involving them (I will not differentiate between axiom and theorem).

For any number a and b if a>b then b-a=inpositivity

inpositivy-a=inpositivity

inpositivity+a is undefined

inpositivity-inpositivity=inpositivity

inpositivity*a=inpositivity

inpositity*inpositivity=positivity

positivity+positivity=positivity

positivity*positivity=positivity

0*positivity=0

0*inpositivity=0

positivity+inpositivity is undefined

positivity - inpositivity = positivity

Call the system defined as the "Dulled natural numbers". Now if you compare this system with integers you might see that the qualities "positive" and "negative" have a close correspondence with these two extra dulled numbers. And this is the intention of the construction. We could consider the dulled system in it's own right. Or we can see it as talking coarsely about the more detailed world of integers.

Note that for expressions explicitly listed as undefined we can find very easily reasons why they don't have locked-in results. For example 5-3=2 and 5-7=-2 their "coarsement" analgoues would be positivity-positivity=positivity and positivity-posititivy=inpositivity. Likewise a rule like "positivity-positivity=0" could take "detailment" forms like "5-7=0" which would be clearly false.

When we are talking about infinities one of the systems we might talk about is the extended real line. It's a similar kind of system in that we add two numbers to the usual real numbers. When there we have expressions like "+infinity - +infinity" being undefined we might not be equiped to have a "detailment" argument to cast it into another system why it is so. And after all if it is good system it must be fine at it's own merits.

But when people read statements involving those kind of infinite numbers they might be reading it as "THE infinity + THE infinity". But there is the option of reading it "AN infinity + AN infinity". In the same way you do not read "THE positivity + THE positivity" but "A positive number + A positive number". When you have expressed that a number is positive you have not yet said how big it is. Likewise when you say that an amount is infinite you have expressed a quality that limits some magnitudes to be out of question but you have not actually specified its magnitude.

As an aside: boolean multiplication and addition of 1*1=1 and 1+1=1 seems to be suspiciuosly similar to positivity multiplication and addition. It might be that boolean algebra is the dulled natural numbers without the positive numbers and inpositivity. Boolean 1 does not translate to natural number 1 and this would be true even if we didn't know about positivity. We could also form other "dulled" number systems such as dulled reals or dulled rationals. Making dulling into an operation that could be applied to arbitrary number systems could be interesting but I don't know the language in which a mathematician could read it as a stand-alone idea.