Is #"infinity" + 1 = "infinity"# ?

Answer 1

Abigail Allen

It depends...

It depends what you mean by "infinity" and by addition if you have such a thing.

Cardinal numbers

The cardinal number of a set of objects is the number of objects in it.

So:

#abs({ 0, 1, 2, 3 }) = abs({"red", "green", "blue", "yellow"}) = 4#

We can express inequality of cardinal numbers in terms of mappings from one set to another.

So, if #A# and #B# are two sets, then:

#abs(A) <= abs(B)#

if and only if there is a one to one function #f:A->B#

For example, we could define:

#{ (f(0) = "red"), (f(1) = "green"), (f(2) = "blue"), (f(3) = "yellow") :}#

So we can tell that:

#abs({ 0, 1, 2, 3 }) <= abs({"red", "green", "blue", "yellow"})#

Similarly, we can find:

#abs({"red", "green", "blue", "yellow"}) <= abs({ 0, 1, 2, 3 })#

and deduce:

#abs({ 0, 1, 2, 3 }) = abs({"red", "green", "blue", "yellow"})#

We can then define the natural numbers (including #0#) as cardinal numbers, where:

#n = abs({ m : m < n })#

So:

#0 = abs(O/}#

#1 = abs({ 0 })#

#2 = abs({ 0, 1 })#

etc.

If we define the natural numbers in this way then if #A# and #B# are disjoint sets, we can define addition as:

#abs(A) + abs(B) = abs(A uu B)#

What happens if you extend this to infinite sets, for example the set of all natural numbers?

Let us write:

#omega = abs( { 0, 1, 2, 3, 4,...} )#

Then we find:

#omega = abs( { 0, 1, 2, 3, 4,... } ) = abs( { 1, 2, 3, 4, 5,...} )#

and:

#omega + 1 = abs( { 1, 2, 3, 4, 5,...} uu { 0 } ) = abs({0, 1, 2, 3, 4, 5,...}) = omega#

So in cardinal addition, we have:

#omega + 1 = omega#

There are other transfinite (i.e. infinite) cardinals larger than #omega#

Ordinal numbers

Ordinal numbers are similar to cardinal numbers, but different. They identify distinct well orderings of sets.

In ordinal arithmetic:

#1 + omega = omega#

but:

#omega + 1 > omega#

Why?

In ordinal arithmetic #omega# is the ordinal number of the natural numbers, which have a natural well ordering:

#0, 1, 2, 3, 4,...#

If we add another number to the beginning of this sequence, then we get a sequence isomorphic to the first:

#k, 0, 1, 2, 3, 4,...# #uarruarruarruarruarr# #darrdarrdarrdarrdarr# #0, 1, 2, 3, 4, 5,...#

but if we add another item to the end of the infinite list, then we get a distinct well ordering:

#0, 1, 2, 3, 4,... k#

So #omega+1 != omega#

Other systems

There are other methods of adding transfinite quantities to ordinary numbers and still getting some kind of addition, etc. to work.

The most extreme is probably Conway's Surreal Numbers.

In that system, all of the numbers #omega - 1#, #omega# and #omega+1# are distinct.

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Answer 2

Savannah Anderson

No, "infinity" + 1 is still "infinity". In mathematics, infinity represents an unbounded quantity, so adding 1 to infinity doesn't change its value; it remains infinite.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.