# Field axioms - is there an axiom for multiplication with zero?

• musicgold
In summary, the conversation discusses the role of axioms in justifying steps in a proof and the possibility of stating the axioms of ##\mathbb{R}## in a way that includes the property ##0\cdot x=0##. It is mentioned that this property can be derived from other axioms and that listing every possible truth about the real number system is not feasible. A link to an article discussing the concept of a set of all truths is also provided.
musicgold
Homework Statement
This is not a homework problem.
I am reading a proof and not sure why a particular step is taken / not taken.
Relevant Equations
why is there no axiom like X . 0 = 0
Please refer to the screenshot below. Every step is justified with an axiom. Please see the link to the origal document at the bottom.

I am trying to understand why the proof was not stopped at the encircled step.
1. Is there no axiom that says ## x \cdot 0 = 0 ## ?
2. Isn't the sixth step using the fact that a 0 can be represented by ## x \cdot 0 ## ?

Document

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The line you marked is just using 1+(-1)=0, replacing the left hand side by the right hand side in the previous line.

The sixth step is using the fact that ##a + (-a)=0##, with ##a = x\cdot0##. Afaik, ##0## only appears in the axioms when defining addition, that it has any special role for multiplication needs to be derived.

musicgold and mfb
##x.0=x.(0+0)= x.0+x.0##
Add ##-x.0## to both sides to obtain ##x.0=0##.

musicgold
I have sometime wondered whether it would be possible to state the axioms of ##\mathbb{R}## in a different way so that ##0\cdot x=0## would be one of them but they would still describe exactly the same number system. The normal set of axioms doesn't contain that property, it has to be derived.

You can't start listing every possible truth about the real number system, anyway, because there's an infinite number of those truths. Actually, it's too infinite to even form a proper set (in the sense of set theory), as the Patrick Grim's proof for "There is no set of all truths" should also apply to any statement "There is no set of all truths about entity X".

http://www.pgrim.org/articles/grim_no_set_of_all_truths.pdf

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musicgold
There is no "axiom" that x.0= 0 because it can be proven from the other axioms. By "distribution", x(y+ 0)= xy+ x0. But y+ 0= y so x(y+ 0)= xy. From xy= xy+ x0, add the "additive inverse" of xy to both sides to get x0= 0.

musicgold
hilbert2 said:
I have sometime wondered whether it would be possible to state the axioms of ##\mathbb{R}## in a different way so that ##0\cdot x=0## would be one of them but they would still describe exactly the same number system. The normal set of axioms doesn't contain that property, it has to be derived.

You can't start listing every possible truth about the real number system, anyway, because there's an infinite number of those truths. Actually, it's too infinite to even form a proper set (in the sense of set theory), as the Patrick Grim's proof for "There is no set of all truths" should also apply to any statement "There is no set of all truths about entity X".

http://www.pgrim.org/articles/grim_no_set_of_all_truths.pdf
In my experience this is called the full theory,aka, all statements /wffs that can be assigned a truth value.

## 1. What are field axioms?

Field axioms are a set of mathematical rules that define the properties and operations of a field. A field is a mathematical structure that contains two operations, addition and multiplication, which follow specific properties.

## 2. What is an axiom for multiplication with zero?

The axiom for multiplication with zero is known as the "zero property of multiplication." It states that any number multiplied by zero will result in a product of zero. This means that zero is an identity element for multiplication.

## 3. Why is there a need for an axiom for multiplication with zero?

The axiom for multiplication with zero is necessary in order to maintain consistency and coherence within the field axioms. Without it, there would be no clear definition of the behavior of zero under multiplication, which could lead to contradictions and inconsistencies within mathematical proofs and equations.

## 4. Can the axiom for multiplication with zero be proven?

No, the axiom for multiplication with zero cannot be proven. Axioms are considered to be self-evident truths and are accepted as starting points for mathematical reasoning. They cannot be proven or derived from other axioms or definitions.

## 5. How does the axiom for multiplication with zero relate to real-world applications?

The axiom for multiplication with zero is essential in various real-world applications, such as in physics, engineering, and economics. For example, in physics, the zero property of multiplication allows for the calculation of work done by a force when the displacement is zero. In economics, it is used in calculating the cost of producing zero units of a good or service.

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