14.23. Answer: 686 numbers. First, we will cross out from the set of numbers $1,2, \ldots, 999$ the numbers that are multiples of 5; their quantity is $\left[\frac{999}{5}\right]=199$. Then, from the same set of numbers $1,2, \ldots, 999$, we will cross out the numbers that are multiples of 7; their quantity is $\left[\frac{999}{7}\right]=142$. In this process, numbers that are multiples of 35 will be crossed out twice. Their quantity is $\left[\frac{999}{35}\right]=28$. Therefore, we have crossed out a total of $199+142-28=313$ numbers, and $999-313=686$ numbers remain.