There are exactly FIVE logical possibilities (setting aside what are or are not Scriptural possibilities) ...
- Single Predestination
- God chooses to save all
- God chooses to damn all
- God chooses some to be saved and the rest can save themselves by their works.
- God chooses some to be damned and the rest can save themselves by their works.
This is a good starting point for a discussion on possibilities. I'd say you've mixed up the concept of predestination and the process of salvation there - i'll explain why and we can work from there.
Predestination is to decree your end destiny or destination pre- anyone's good or evil. And logically there are only 2 destinies - life or death. So again logically, there are only 3 options for God to choose over a person -
1. God decrees end destiny of life before any of his good or evil.
2. God decrees end destiny of death before any of his good or evil.
3. God does not decree anything before any of his good or evil, passing over and reserving to decree after his good or evil.
Would you agree with this so far, just from a logical standpoint?
If so, moving further to the process of fulfilling such predestination in a person -
1. Predestined or Promised Salvation requires God to show unconditional mercy and give a new heart and mind as well as to rebirth him in a new nature so as to assuredly save him.
2. Predestined Condemnation requires God to, at minimum, do nothing at all for he will work out condemnation for himself anyway apart from God working in him.
3. No Predestination of either allows for God to show conditional mercy to him in giving him a new heart and mind, not
assuredly saving him by birthing him in a new nature (since God's made no such Promise to do so as with Predestined Salvation above), but providing him the means of Salvation through faith in Christ alone which he must self-determine to endure in to the end. This is to prove that all flesh is unprofitable even when the means for salvation are provided in all sincerity.
Single Predestination holds the above 1. for all the elect and the above 3. for all the non-elect. Where is the logical or Scriptural fallacy - in fact 3. above factors in the Hebrews falling away passages unlike calvinism.
Can a man be saved by his own efforts apart from God?
This is the flaw in your framing the question - Single Predestination never implied that the non-elect are saved, it only says God didn't decree his condemnation pre- his works of good or evil. Do you see the precise nuance here?