Sherlock Holmes gets all the glory. Some say Watson deserves half, but even that isn't the full story. Far less famous are the Baker Street Irregulars, a band of street urchins paid by Holmes to act as his eyes-and-ears on cases. They each get a shilling a day plus expenses, with rewards for vital clues: pocket change for Holmes, but they prove invaluable on multiple occasions. Yet for all this, few people outside the Sherlock Holmes fanbase, and even fewer within the Sherlock Holmes canon, even know of their existence. Everyone knows Sherlock Holmes as the mastermind, the ultimate detective who miraculously knows all and sees all, and even what he cannot know or see he somehow deduces. No one sees his informants.

There are limits to how much the humanoid mind can pay attention to, and even magic cannot break them. Divinations can only reveal so much, and this is the major problem with a single mage taking over the duties of an entire law enforcement agency. Even if a mage can be there to stop all crime that occurs, he must first know all crime that occurs, and divinations are of limited use in this.

But by partnering with the "invisible" people of the city to be his eyes-and-ears -people that nobody sees because everyone looks away- a single wizard might seem to be working alone. To further the question's analogy with ancient Carthage, if the wizard says his "Elephants" tell him about crimes, people will assume he's talking about constructs or summoned outsiders or familiars, but almost no one will suspect the truth. No one ever does.

No.

If you intend your wizard to effectively replace an entire police force, then probably not. If you intend for your wizard to effectively stop crime, then absolutely not.

If we take crimes committed yearly per 100,000 people in the United States (just for ease of data gathering—your particular city might be more or less naturally law-abiding, but we have to pick some baseline), your crimes committed yearly would be:

Total violent crime: 1,100 Total property crime: 7,500

I suspect given amount the violence in the world supported by the D&D rule set, the incidence of crime would be much, much higher. Much of the ruleset supports killing things and taking stuff.

That is 8,600 crimes per year, or about 23 per day. Most of these would have no witnesses (most property crimes don't), or no way to report to authorities in time to prevent (peasants don't have telephones). It also doesn't address "victimless crimes" such as loitering, public urination, being a public nuisance, bribery, contract breaking, etc.

The onus of investigation and enforcement would fall upon one wizard with a limited number of spell slots and breadth of attention. He also has to sleep (or he won't get spells back, for one thing), which means that every day will have 8 or so hours of no enforcement to speak of.

There is only one way that he might reasonable reduce crime, and that is deterrence. If he were to institute draconic disproportionate response to crime, that might make a dent in the amount. People might be less inclined to commit petty crimes if the punishment for loitering was slow and painful public execution (which the wizard would have to handle personally, since he is the entire police force.) The wizard would also have to punish crimes of diminishing importance (ballooning the perceived crime rate) in order to make the point. People being people, they would live in fear of being unjustly accused, and many would view even just accusations as unfair, cruel or inhuman. This would foment resentment, rebellion and hostility.

Such a wizard would not be a law enforcer. He would be a tyrant.

The answer is probably yes.
What is the standard of law that a normal city would enjoy in your world? It is maybe a silly question, but threats and bribes are pretty normal if you ask me so there is crime in normal city and that is the reason why it is not necessary for the wizard to be everywhere and punish every criminal for every crime.

But I think that there will be difference in what crimes will be solved. Generally speaking for one experienced person it is easier to solve few big crimes done by more powerful criminals. For many low level guards it is easier to solve great number of small crimes done by low level criminals.

How many guards you have in one patrol? What are their levels and classes? If you have four one level fighters what they can do against one six level rogue, or wizard? If you have one seventeen level wizard what they can do against cleverly synchronised twenty crimes at once across the whole city?

Is it true, that there will be more low level crimes with your wizard? Not necessarily. He will probably have more respect and looks more intimidatingly for the low levels.

At the end. The result depends on your world. It depends on the severity of penalties, but I believe that in some settings it is definitely possible.

Yes

Certainly at high levels this is trivial. A 17th level wizard has access to the spell Time Stop, which lets him stop time, and Plane Shift, which lets him travel to the Astral Plane so as to better stop time. Such a wizard may investigate and/or fight crime wheresoever he chooses for as long as he wishes-- he is limited only by his patience. If he wanted he could systematically use Stone Tell via Wish ( with material components acquired via Fabricate) to literally leave no stone unturned. The primary problem with this method, of course, is that it is very boring and very lonely. Nonetheless, it is enough to stop crime, like, all of it.

Before 17th level, the Wizard is significantly less omnipotent. Nonetheless, there exist several spells that seem designed specifically to let you do what you want. For example, at 15th level there is the spell Greater Prying Eyes, which allows the wizard to send a small army of better-than-invisible programmed sentinels that know everything the wizard knows with 120 foot True Seeing effects and a +25 Perception modifier to police the town for him. Obviously the response time to petty crime in such a city would be kinda slow-- the eyes can only travel at human running speeds, and there's not much reason to have them return immediately when they see a petty crime so it'd probably have to wait till the end of the day-- but it'd be much more thorough system of monitoring than a well-staffed mundane police force. More pressing crimes could be dealt with more rapidly, either by virtue of the premature return of the eye that found the crime, or by citizens raising an alarm to call the wizard's attention, just as with a traditional police force.

Contingency is also available at 15th level, which helps out by letting catch criminals with something like 'when a crime is comitted...' and then any of several dozen spells that affect the wizard while also letting them find the criminals.

Before 15th level, only the regular Prying Eyes are available, and they are much less effective guardsmen, lacking darkvision or any way of seeing past skilled stealth attempts. They are reasonable substitutes for low-level human constables, but lack the physical power of such a guard, and are no substitute for more elite mundane guardsmen. Nonetheless, at 13th level, the spell Simulacrum allows the wizard to create permanent non-magical illusory duplicates of himself with half his class levels that are under his absolute control. At level 14 these duplicates are 7th level casters capable of 4th level spell, while at level 13 they are 6th level casters capable of only 3rd level spells. Nonetheless, they are reasonable servitors with which to augment the eyes, costing only 500 gp per level. If the wizard is willing to use servitors that don't share his likeness but are nonetheless under his absolute control, he can instead create casters of his level by duplicating a caster of twice his level, partial casters, like a Conqueror Worm, Omnipath, or Wyrm Dragon, or non-casters, such as a Hekatonkheires. Such servitors cost approximately 13000 gp each, but are well worth the expense. Note that duplicates of evil creatures are not themselves evil.

Contingency may also be useful here-- tying it to a Scrying spell (which the caster qualifies for at 12th level) allows the Wizard to scry upon the first person to commit a crime in the city after the casting of contingency (or perhaps a specific category of crimes, or crimes beyond a certain severity, or something).

Before 13th level, things are much harder. Prying Eyes, the weaker version, is available at 9th level, as is Contact Other Plane, but with a cult of seventh level Clerics as a reasonably possible criminal situation in the city, the Wizard can't really handle all the potential crime on his own.

That puts our minimum at 13th level, I think, though 9th level may be possible with skilled play.

P.S. Technically you can do this at level 7 with armies of Undead, since being good aligned doesn't stop a wizard from casting evil spells, since he's not a Cleric (he can make up for it with Atonement, or by casting Summon Monster for a good creature). You can also find most of the crime at level 3 with Commune with Birds, but a 3rd level wizard on his own isn't much of a match for a 7th level wizard with leadership.

Lastly, you can do this at level 1 if 'I bought all the sheep ever, and now they are the police' works in your game. If you're a gnomish wizard you can even possibly talk to your cheapo-army of constable-rats. Unfortunately animal police forces are not particularly popular with most polities, nor are they likely to be particularly loyal, given that your Diplomacy skill is probably not that great at first level. Also, it won't look like you're doing it single-handedly, so it won't count.

Its very reasonable to assume that D&D guards would be no more effective than modern officers. This assumption allows us to establish a baseline answer based off modern law-enforcement numbers. Looking here, it seems that at minimum, there's 15.9 Officers per 10k citizens. That's 477 modern Officers to police your example city.

To estimate if a singular Wizard of level 7 could compensate for 477 Officers, let's use the most standard measure of effectiveness in Pathfinder: Challenge Rating. A level 7 Wizard can, by CR, take the place of 2 level 5 Wizards, 4 level 3 Wizards, or 8 level 1 Wizards.

Even on the ridiculously generous assumption that a level 1 Wizard is more effective than 10 modern Officers, our level 7 Wizard is still only doing the job of 80 modern Officers, of the required 477 to fully protect this city.

The assumptions I laid out, above, don't properly assess exactly how inadequate a level 7 Wizard would be, but the assumptions were all in the Wizard's favor, so closer examination would not help the Wizard's case.

One commenter opined about the level of a Wizard who would be able to replicate the guardsman. To that end, I shall reel in my constraints to more plausible values.

First, let's examine the CR of a modern Officer. I previously bid 10 officers as CR 1, which I accurately called ridiculous. More plausibly, these trained Officers are at least level 2 in a PC class, being highly changed and FAR above the fighting and investigating talents of the average person. Thus, 1 Officer being CR 2 is far more plausible, albeit still an assumption I feel weighs below our Officer's talent. At that point, only a level 20 Wizard could match the CR of our Officers.

Then, we're forced to ask, does even a level 20 Wizard, without permanent aids, have the ability to work the shifts required? And the answer is no. Officers are called upon to work both day and night, often in mandatorily in multiple places at the same time. A singular wizard, even of level 20, could not keep up without the kinds of cheese that would render this question moot. He'd be unable to prep spells due to working 24/7. Most of his most potent spells cannot be brought to bear against the some-3816 problems Officers would face on a daily basis.

No, no Wizard, prior to Epic, could possibly hope to do what our fine Officers do, by himself without cheese or minion-mancy.

I believe the answer would be no, and here's why:

I firmly agree with the statement "No major crime committed would go unpunished." Major crimes being public manslaughter, openly aggressive acts of vandalism/robbery, kidnappings, or the like. This is due to two factors. 1: Most persons capable of committing these crimes would likely realize who and what they'd be facing, and so a good number of them simply wouldn't commit said crimes. 2: the wizard does what a wizard does.

I do not agree with said wizard being able to enforce the law the way a fully staffed guard corps would be able to, as there are an unending parade of petty crimes happening that he'd never be able to keep up with. He's a powerful wizard, but he's not omniscient (while he may have scrying magic, they don't last forever, and they don't cover an entire city OR able to see everything clearly at once) and he's not omnipresent (while he'd likely have teleportation magic, he doesn't have an endless number of spell slots to use them).

I believe he'd do great things in dropping a city's crime rate (Khelben Blackstaff did wonders reducing Waterdeeps magical disturbances when he moved in, for instance) but I don't think he'd be able to replace the guards. Too much ground, and not able to keep up with "my neighbor kicked my dog" and "but she let her dog in my yard first!" type pettiness."

Smallest city is hard to determine, but I'd say a few thousand or so. A few thousand citizens is enough that crime rate is low enough to personally intervene and be able to replace the guards, but the citizenry is also probably docile enough to not have much crime anyway.