Infinity: Limitless or Limited? Exploring the Concept of Infinity in Mathematics

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In summary: So I say, "Ah, but what about all the numbers between 1 and 2?"Your response: "I have 1, 2, 3, 4, 5, 6, 7, 8... and 1.5."Do you see what I'm trying to say?In summary, the concept of infinity can be confusing, but it is important to understand that there are different sizes of infinities and not all infinities are the same. The number of elements in an interval, such as (0,1) or (0,2
  • #141
If I have a sphere and I remove a point from it, it still has volume right? But topologically speaking a sphere with a point removed is the same as a plane, so where does the volume go when I stretch the sphere into a plane?
 
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  • #142
gravenewworld said:
If I have a sphere and I remove a point from it, it still has volume right? But topologically speaking a sphere with a point removed is the same as a plane, so where does the volume go when I stretch the sphere into a plane?
Do you mean a sphere or a ball?
If you mean a ball then the volume of the ball and the volume of a ball with one point removed (whatever you mean by removing a point) is the same. :smile:
Remember that a point does not occupy any space!

If you are really talking about a sphere then well we just learned that a sphere has no volume just like a circle has no area.

And by the way removing a point from a sphere does not reduce the area of the object.
 
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  • #143
gravenewworld said:
If I have a sphere and I remove a point from it, it still has volume right? But topologically speaking a sphere with a point removed is the same as a plane, so where does the volume go when I stretch the sphere into a plane?

Ok firstly a sphere has zero volume (a sphere is a closed subset of R^3 and every closed subset is measurable under the normal legesgue measure). However volume is not a topological invariant (think length of (0, 1) which is homeomorphic to the entire real line) but is rather invariant under isometries (ie. rigid motions) which continuous functions need not be. Remember that not even angles need to be preserved under continuous mappings, so volumes being preserved is very special indeed.
 
  • #144
river_rat said:
Ok firstly a sphere has zero volume (a sphere is a closed subset of R^3 and every closed subset is measurable under the normal legesgue measure). However volume is not a topological invariant (think length of (0, 1) which is homeomorphic to the entire real line) but is rather invariant under isometries (ie. rigid motions) which continuous functions need not be. Remember that not even angles need to be preserved under continuous mappings, so volumes being preserved is very special indeed.
lol sounds like i need to stick to logic and algebra. analysis and topology were always my weakness.
 
<h2>1. What is infinity?</h2><p>Infinity is a concept in mathematics that represents something that is unbounded, unlimited, or never-ending. It is often symbolized by the symbol ∞.</p><h2>2. Is infinity a number?</h2><p>No, infinity is not a number. It is a concept that represents something that is without limit or bound.</p><h2>3. Can infinity be measured?</h2><p>No, infinity cannot be measured because it is not a tangible quantity. It is a concept that represents something that is limitless.</p><h2>4. Is infinity the biggest number?</h2><p>No, infinity is not a number and therefore cannot be compared to other numbers. It is a concept that represents something that is unbounded.</p><h2>5. How is infinity used in mathematics?</h2><p>Infinity is used in various areas of mathematics, such as calculus, set theory, and geometry. It is often used to represent a limit or an unbounded quantity in equations and formulas. It is also used in the concept of infinite sets and the idea of approaching infinity in mathematical calculations.</p>

1. What is infinity?

Infinity is a concept in mathematics that represents something that is unbounded, unlimited, or never-ending. It is often symbolized by the symbol ∞.

2. Is infinity a number?

No, infinity is not a number. It is a concept that represents something that is without limit or bound.

3. Can infinity be measured?

No, infinity cannot be measured because it is not a tangible quantity. It is a concept that represents something that is limitless.

4. Is infinity the biggest number?

No, infinity is not a number and therefore cannot be compared to other numbers. It is a concept that represents something that is unbounded.

5. How is infinity used in mathematics?

Infinity is used in various areas of mathematics, such as calculus, set theory, and geometry. It is often used to represent a limit or an unbounded quantity in equations and formulas. It is also used in the concept of infinite sets and the idea of approaching infinity in mathematical calculations.

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