Are there points on a plane that are an infinite distance from the origin (0,0)? ?

In real numbers, there is not.

The concept of an upper bound is a finite number that is bigger than the measure you are seeking.

What is the largest distance from (0, 0) on the plane? Whatever finite number you propose as an upper bound for that largest distance, we can find a coordinate (using finite numbers) that is further away.  Therefore, there is "no finite" bound. That is what we call "infinite" (our boundless) in this context.

"infinite" is not, itself, a number.

In the Complex Field (which is not well-ordered), the concept of using "infinity" is needed, and it is given a direction. However, this is done simply to allow closure of the field (needed for the application of certain concepts), and the direction was simply "chosen" by convention (there is no reason why it is that direction rather than another).

In a right-angle triangle, if one side is kept at a fixed length and you keep extending the other side "without bound", then the length of the hypotenuse (always longer than any of the other two sides) will tend to be closer and closer to the side being extended.

At the limit (at "infinity"), the hypotenuse would be the same length as the long side. That is because the short side, in relation, would by then represent exactly 0% of the long side, making the whole figure a degenerate triangle (angles of 90, 90 and 0).

Such a triangle (with two infinite sides) does not exist. It is only a "tendency" as the values "tend" to infinity.

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When infinities are counting the same things, they are of equal size.

Take, for example, the line of numbers:

N = 0, 1, 2, 3, 4, 5.... forever.

it contains "infinitely many" numbers. Whatever number you could propose as being the largest, we can simply add 1 to it and get a new valid and finite number (this was understood at least 2000 years ago).

Now, lets create a new list by doubling all the numbers in N; let's call it "EVEN"

EVEN = 0, 2, 4, 6, 8, 10... forever.

By the way we built it (we say "by construction"), it must have exactly the same number of numbers as "N".

Let's create yet another list by adding 1 to all numbers in EVEN; this new list is called "ODD"

ODD = 1, 3, 5, 7, 9, 11... forever

By construction, ODD has the same number of numbers as EVEN; therefore, it has the same number of numbers as N.

It is easy to prove that none of the numbers in EVEN are found in ODD (and vice versa). They are two totally distinct lists (there is no duplication).

NEW = EVEN + ODD

NEW = 0, 1, 2, 3, 4, 5... forever.

By construction (if you could count infinities), you would expect NEW to be twice as long as "N".

And yet, they are EXACTLY the same. Otherwise, you should be able to find some numbers in "NEW" that were not already in "N".

A point in the Cartesian plane is an ordered pair (x,y) of real numbers.  Infinity is not a real number, so the distance between any two points on such a plane is a non-negative real number.  There are no infinite distances.

If you extend such a system to include values other than pairs of real numbers as "points", then you could also extend the definition of distance to allow distances that are not real numbers.  It's up to you to show that the resulting system still acts like a geometry. Can you still form a unique line through any two points?  Can you define a circle centered at one of those extra points?  Can you bisect a line segment when one of the endpoints is one of those new "points"?  Etc.

No. Every point is a finite distance away. But the plane is unbounded. For any number you can name, there are points whose distance is greater that that number.

The answer to this is related to the numbers themselves. While there is no largest number, infinity is not a number. Infinity is just a concept, not an actual number.

The issue is does a true plane exist or is it just a flawed mathematical concept.

If all planes are actually a sphere whose radius is such that at some point all lines return to their point of origin, then infinity is 1/2 of the total circumference (the point equal distant from the starting point). This makes zero twice infinity because that number represents the total distance traveled, or if you prefer zero is twice infinity. Therefore infinity is zero. The point you are looking for is equal to 0/2.