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I am not able to understand these axiom and statements by hilbert.

" theorem 8. If two straight lines a, b of a plane do not meet a third straight line c of the same plane, then they do not meet each other.

For, if a, b had a point A in common, there would then exist in the same plane with c two straight lines a and b each passing through the point A and not meeting the straight line c. This condition of affairs is, however, contradictory to the second assertion contained in the axiom of parallels as originally stated. Conversely, the second part of the axiom of parallels, in its original form, follows as a consequence of theorem 8."

Please explain what he's trying to say ?

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  • $\begingroup$ "contradictory to the second assertion contained in the axiom of parallels as originally stated" ... You'd have to go back and re-read the axiom of parallels to find out what he means by this, now, wouldn't you? $\endgroup$
    – David K
    Nov 22, 2021 at 3:06
  • $\begingroup$ Just draw the figure! It will be obvious. $\endgroup$ Nov 22, 2021 at 3:09

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"Two straight lines do not meet in a plane" means that they are parallel. What he's saying here is that if $a$ and $c$ are parallel and $b$ and $c$ are parallel, then $a$ and $b$ are parallel.

His proof uses what I assume is commonly known as the parallel postulate: given a line and a point not on that line, there exists a unique line through the point that is parallel to the original line. Hilbert is observing that there's a contradiction if you consider $c$ to be the line and the intersection of $a$ and $b$ to be the point in the premise of the parallel postulate. There are now two lines (namely $a$ and $b$) both of which pass through the point and both of which are parallel to $c$. This contradiction means that $a$ and $b$ must have in fact not intersected at all.

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